phd in abstract algebra

Course Catalog

• Mathematics, PhD

for the Doctor of Philosophy in Mathematics

Graduate Degree Programs in Mathematics

• Actuarial Science, MS
• Applied Mathematics, MS
• Mathematics, MS
• Predictive Analytics and Risk Management, MS
• Enterprise Risk Management
• Financial and Insurance Analytics
• Actuarial Science & Risk Analytics
• Computational Science and Engineering
• Teaching of Mathematics, MS

Students working toward a Ph.D. degree usually require from four to six years to complete the requirements. Each student must pass the comprehensive examinations (testing the student's knowledge of basic graduate-level mathematics in algebra, analysis, and other areas) and the preliminary examination (testing the student's ability to begin or continue research in a chosen field). Students must also write and defend a research thesis in their field of mathematics.

For additional details and requirements refer to the department's  Guide to Graduate Studies  and the  Graduate College Handbook . 

Requirements

Other requirements .

• Acquire a foundation in abstract algebra at the graduate level.
• Acquire a foundation in real analysis at the graduate level.
• Acquire a suitable breadth of knowledge to provide a foundation for undertaking high-level research.
• Gain a broad understanding of the range of current research in the mathematical sciences.
• Demonstrate depth of knowledge in chosen area of research specialization.
• Gain the ability to conduct independent mathematical research at a professional level.
• Gain experience and competence in the teaching of mathematics at the college level.

Mathematics Department Department Chair: Vera Hur Director of Graduate Studies: Yuliy Baryshnikov Mathematics Department website Mathematics Department  faculty Mathematics faculty research  273 Altgeld Hall, 1409 West Green Street, Urbana, IL 61801   (217) 333-5749 Mathematics email

College of Liberal Arts & Sciences College of Liberal Arts & Sciences website

Admissions Mathematics Admissions & Requirements Graduate College Admissions & Requirements

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Doctor of Philosophy (PhD)

Program synopsis and training.

 The Doctor of Philosophy (PhD) in mathematics is the highest degree offered by our program. Graduates will have demonstrated their ability to conduct independent scientific research and contribute new mathematical knowledge and scholarship in their area of specialization. They will be well-supported and well prepared for research and faculty positions at academic institutions anywhere in the world. Owing to their independence, analytic abilities, and proven tenacity, our PhD graduates are also sought after by private and government employers.

Our PhD program offers two tracks, one for Theoretical Mathematics and one for Applied Mathematics . The tracks differ only in the course and  qualifying requirements during the first two years. Applicants are required to decide on one of the tracks and applications will be evaluated subject to respective criteria described below.

Once students have passed their Qualifying Requirements, the two tracks merge and there is no distinction in later examinations and research opportunities. In particular, the candidacy exam for both tracks consists of a research proposal, the graduate faculty available for advising is the same, and the final degree and thesis defense are independent of the initially chosen track.

Expected Preparations for Admission

Competitive applicants to the theoretical track are expected to have strong foundations in Real Analysis and Abstract Algebra, equivalent to our Math 5201 - 5202  and Math 5111 - 5112 sequences.

Expected preparations for the applied track include the equivalents of a rigorous Real Analysis course (such as Math 5201 ), a strong background in Linear Algebra, as well as an introductory course in Scientific Computing.

Besides these basic requirements, competitive applicants in either track submit evidence for a broad formation in mathematics at the upper-division or beginning graduate level. Relevant coursework in other mathematical or quantitative sciences may also be considered, especially for the applied track.

Prior research experiences are not required for either track, and we routinely admit students without significant research background. Nevertheless, applicants are encouraged to include accounts of research and independent project endeavors as well as letters of supervising mentors in order to be more competitive for fellowship considerations. The research component is likely to have greater weight in applications to the applied track.

These prepared documents serve to provide our admission committee with a narrative overview of the applicant's mathematical trajectory. Their primary focus should, therefore, be to enumerate and describe any evidence of mathematical ability and mathematical promise. The information included in the documents should be well-organized, comprehensive, informative, specific, and relevant. This will help our committee to properly and efficiently evaluate the high number of applications we receive each year.

Our Graduate Recruitment Committee will generally  not  consider GRE test scores for this Autumn 2024 admissions. If you have already taken the test, please do not self-report the scores to us. In exceptional circumstances students may have the option to report unofficially. 

International students whose native language is not English and are not exempt should score at least a 20 on the Speaking portion of the TOEFL or at least 6.5 on the IELTS Speaking portion.  We also recommend an overall score of at least 95 on TOEFL or at least 7.0 on IELTS.  For a list of exempt countries, please see  https://gpadmissions.osu.edu/intl/additional-requirements-to-apply.html

Qualifying Requirements by Track

The qualifying requirements for the theoretical track are fulfilled by passing our Abstract Algebra course sequence  ( Math 6111 , Math 6112 ) and  our Real Analysis course sequence ( Math 6211 , Math 6212 ), each with at least an A-, or  by passing a respective examination.

The qualifying requirements for the applied track combine a mandatory Scientific Computing course ( Math 6601 ), one of the algebra or analysis courses, and three additional courses chosen from  Math 6602 , Math 6411 ,  Math 6451 , and the courses comprising the algebra and analysis sequences.

The breadth requirements in the applied track are more flexible than in the theoretical track, but also include a mandatory graduate course in a non-math STEM department from an approved list. 

You can find more information about our PhD program requirement here .

Opportunities & Outcomes

The research opportunities and academic outcomes of our doctoral program are described in detail in the Graduate Program Prospectus  [pdf]. 

Our department has about 80 active graduate faculty on the Columbus and regional campuses. Virtually every area of mathematics is represented in our program, with a sampling displayed below.

• Commutative, Non-commutative, & Quantum Algebra
• Analytic, Algebraic, Computational Number Theory
• Algebraic Geometry, Tropical Geometry
• Applied Mathematics, Mathematical Physics
• Real and Complex Analysis
• Functional Analysis, Operator Algebras
• Combinatorics and Graph Theory
• Differential Geometry
• Dynamical Systems and Ergodic Theory
• Financial and Actuarial Mathematics
• Logic and Foundations
• Probability Theory, Statistical Mechanics
• Mathematical Biology
• Ordinary and Partial Differential Equations
• Representation theory
• Scientific Computing
• Topology, Topological Data Analysis

See also our  Applied Mathematics Topics List  [pdf].

Our program offers many support opportunities without teaching duties as well, to allow more time for scientific endeavors. These opportunities include university fellowships, external funding, and departmental fellowships and special assignments. See the  Financial Support  page for more details.

The median time to degree completion in our program is below six years but also varies significantly among our students, with as little as four years for students entering with substantial prior preparations. Funding is guaranteed for six years and can be extended to seven years with advisor support and the permission of the Graduate Studies Committee.  

Most of our graduates continue their careers in academia. Post-doctoral placements in the last two years include, for example, UCLA, Stanford, ETH-Zürich, Brown University, University of Michigan, Northwestern University, University of Vienna, EPF Lausanne, Free University at Berlin, Purdue University, and University of Utah. In recent years our graduates also went to Princeton University, IAS, University of Chicago, Yale University, University of Michigan, Cal-Tech, Northwestern University, University of Texas, Duke University, SUNY Stony Brook, Purdue University, University of North Carolina - Chapel Hill, and Indiana University. Recent non-academic placements include Google, Facebook, Amazon, NSA, and prestigious financial institutions.

Students also have access to training and networking opportunities that prepare them better for careers in private industry and teaching - for example, through the Erdős Institute  - and are regularly offered highly competitive positions in the industry. 

Nearly half of the graduate population consists of domestic students coming from both larger universities and smaller liberal arts colleges with a solid math curriculum. And as a program group member of the National Math Alliance , we are dedicated to enhancing diversity in our program and the scientific community. The International students in our program come from all parts of the world with a wide variety of educational backgrounds.

Prospective students:  [email protected]

Graduate Office Department of Mathematics The Ohio State University 231 W 18th Avenue ( MA 102 ) Columbus, Ohio 43210 United States of America                

Phone: (614) 292-6274 Fax: (614) 292-1479

[pdf] - Some links on this page are to .pdf files. If you need these files in a more accessible format, please email  [email protected] . PDF files require the use of Adobe Acrobat Reader software to open them. If you do not have Reader, you may use the following link to Adobe to download it for free at:  Adobe Acrobat Reader .

Mathematics, PHD

On this page:, at a glance: program details.

• Location: Tempe campus
• Second Language Requirement: No

Program Description

Degree Awarded: PHD Mathematics

The PhD program in mathematics is intended for students with exceptional mathematical ability. The program emphasizes a solid mathematical foundation and promotes innovative scholarship in mathematics and its many related disciplines.

The School of Mathematical and Statistical Sciences has very active research groups in analysis, number theory, geometry and discrete mathematics.

Degree Requirements

84 credit hours, a written comprehensive exam, a prospectus and a dissertation

Required Core (3 credit hours) MAT 501 Geometry and Topology of Manifolds I (3) or MAT 516 Graph Theory I (3) or MAT 543 Abstract Algebra I (3) or MAT 570 Real Analysis I (3)

Other Requirements (3 credit hours) MAT 591 Seminar (3)

Electives (24-39 credit hours)

Research (27-42 credit hours) MAT 792 Research

Culminating Experience (12 credit hours) MAT 799 Dissertation (12)

Additional Curriculum Information Electives are to be chosen from math or related area courses approved by the student's supervisory committee.

Students must pass:

• two qualifying examinations
• a written comprehensive examination
• an oral dissertation prospectus defense

Students should see the department website for examination information.

Each student must write a dissertation and defend it orally in front of five dissertation committee members.

Admission Requirements

Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and Sciences.

Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics or a closely related area from a regionally accredited institution.

Applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in the last 60 hours of their first bachelor's degree program or a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in an applicable master's degree program.

All applicants must submit:

• graduate admission application and application fee
• official transcripts
• statement of education and career goals
• three letters of recommendation
• proof of English proficiency

Additional Application Information An applicant whose native language is not English must provide proof of English proficiency regardless of their current residency.

Additional eligibility requirements include competitiveness in an applicant pool as evidenced by coursework in linear algebra (equivalent to ASU course MAT 342 or MAT 343) and advanced calculus (equivalent to ASU course MAT 371), and it is desirable that applicants have scientific programming skills.

Next Steps to attend ASU

Learn about our programs, apply to a program, visit our campus, application deadlines, learning outcomes.

• Address an original research question in mathematics.
• Able to complete original research in theoretical mathematics.
• Apply advanced mathematical skills in coursework and research.

Career Opportunities

Graduates of the doctoral program in mathematics possess sophisticated mathematical skills required for careers in many different sectors, including education, industry and government. Potential career opportunities include:

• faculty-track academic
• finance and investment analyst
• mathematician
• mathematics professor, instructor or researcher
• operations research analyst
• statistician

Program Contact Information

If you have questions related to admission, please click here to request information and an admission specialist will reach out to you directly. For questions regarding faculty or courses, please use the contact information below.

• [email protected]
• 480/965-3951

• Schools & departments

Algebra PhD

Awards: PhD

Study modes: Full-time, Part-time

Funding opportunities

Programme website: Algebra

Upcoming Introduction to Postgraduate Study and Research events

Join us online on the 19th June or 26th June to learn more about studying and researching at Edinburgh.

Choose your event and register

Research profile

Our research group has strong links with the Geometry and Topology group, as well as Mathematical Physics. You’ll find these invaluable as opportunities to discuss your work and expand your thinking.

The School of Mathematics is a vibrant community of more than 100 academic and related staff supervising over 100 PhD students.

Working within one of the largest mathematics groups in the UK, you’ll be completing your degree in an environment that hums with a busy graduate school life, and you’ll have the chance to make your mark in seminars, workshops, clubs and outings.

The Hodge Institute is the home of algebra, geometry, number theory and topology research groups in the School of Mathematics.

• The Hodge Institute

Our interests include:

• non-commutative ring theory
• non-commutative algebraic geometry
• the geometry of algebraic numbers
• Lie-theoretic representation theory
• quantum algebra
• category theory

While we offer a large community of researchers under one roof, we believe in encouraging you to gain as broad a perspective as possible. The best way to do this is to involve yourself in the international dialogue on your research area, through attending conferences and symposia, and visiting your peers in centres of research worldwide.

Throughout your studies, you’ll be given opportunities to travel to events and institutions that will allow you to gain this perspective and open up new areas of investigation.

You can find out more on the Maxwell Institute Graduate School site.

Training and support

Mathematics has connections stretching across all the scientific disciplines and beyond. Our School is one of the country’s largest mathematics research communities in its own right, but you will also benefit from Edinburgh’s high-level collaborations, both regional and international.

Research students will have a primary and secondary supervisor and the opportunity to network with a large and varied peer group. You will be carrying out your research in the company of eminent figures and be exposed to a steady stream of distinguished researchers from all over the world.

Our status as one of the most prestigious schools in the UK for mathematics attracts highly respected staff. Many of our 100+ current academics are leaders in their fields and have been recognised with international awards.

Researchers are encouraged to travel and participate in conferences and seminars. You’ll also be in the right place in Edinburgh to meet distinguished researchers from all over the world who are attracted to conferences held at the School and the various collaborative centres based here.

You’ll find opportunities for networking that could have far-reaching effects on your career in mathematics.

You will enjoy excellent facilities, ranging from one of the world’s major supercomputing hubs to generous library provision for research at the leading level, including the new Noreen and Kenneth Murray Library at King’s Buildings.

Students have access to more than 1,400 computers in suites distributed across the University’s sites, many of which are open 24 hours a day. In addition, if you are a research student, you will have your own desk with desktop computer.

We provide all our mathematics postgraduates with access to software packages such as Maple, Matlab and Mathematica. Research students are allocated parallel computing time on ‘Eddie’ – the Edinburgh Compute and Data Facility. It is also possible to arrange use of the BlueGene/Q supercomputer facility if your research requires it.

Advice on applications

For advice on applications, see

• Hodge Institute page

We particularly encourage women and other underrepresented groups in mathematics to apply, and work with groups such as the Piscopia Initiative to improve the representation and inclusion of women and minorities in mathematics.

Entry requirements

These entry requirements are for the 2024/25 academic year and requirements for future academic years may differ. Entry requirements for the 2025/26 academic year will be published on 1 Oct 2024.

A UK first class honours degree, or its international equivalent, in an appropriate subject; or a UK 2:1 honours degree plus a UK masters degree, or their international equivalents; or relevant qualifications and experience.

International qualifications

Check whether your international qualifications meet our general entry requirements:

• Entry requirements by country
• English language requirements

Regardless of your nationality or country of residence, you must demonstrate a level of English language competency at a level that will enable you to succeed in your studies.

English language tests

We accept the following English language qualifications at the grades specified:

• IELTS Academic: total 6.5 with at least 6.0 in each component. We do not accept IELTS One Skill Retake to meet our English language requirements.
• TOEFL-iBT (including Home Edition): total 92 with at least 20 in each component. We do not accept TOEFL MyBest Score to meet our English language requirements.
• C1 Advanced ( CAE ) / C2 Proficiency ( CPE ): total 176 with at least 169 in each component.
• Trinity ISE : ISE II with distinctions in all four components.
• PTE Academic: total 62 with at least 59 in each component.

Your English language qualification must be no more than three and a half years old from the start date of the programme you are applying to study, unless you are using IELTS , TOEFL, Trinity ISE or PTE , in which case it must be no more than two years old.

Degrees taught and assessed in English

We also accept an undergraduate or postgraduate degree that has been taught and assessed in English in a majority English speaking country, as defined by UK Visas and Immigration:

• UKVI list of majority English speaking countries

We also accept a degree that has been taught and assessed in English from a university on our list of approved universities in non-majority English speaking countries (non-MESC).

• Approved universities in non-MESC

If you are not a national of a majority English speaking country, then your degree must be no more than five years old* at the beginning of your programme of study. (*Revised 05 March 2024 to extend degree validity to five years.)

Find out more about our language requirements:

• Academic Technology Approval Scheme

If you are not an EU , EEA or Swiss national, you may need an Academic Technology Approval Scheme clearance certificate in order to study this programme.

Fees and costs

Tuition fees, scholarships and funding, featured funding.

• School of Mathematics funding opportunities
• Research scholarships for international students

UK government postgraduate loans

If you live in the UK, you may be able to apply for a postgraduate loan from one of the UK's governments.

The type and amount of financial support you are eligible for will depend on:

• your programme
• the duration of your studies
• your residency status.

Programmes studied on a part-time intermittent basis are not eligible.

• UK government and other external funding

Other funding opportunities

Search for scholarships and funding opportunities:

• Search for funding

Further information

• Graduate School Administrator
• Phone: +44 (0)131 650 5085
• Contact: [email protected]
• School of Mathematics
• James Clerk Maxwell Building
• Peter Guthrie Tait Road
• The King's Buildings Campus
• Programme: Algebra
• School: Mathematics
• College: Science & Engineering

Select your programme and preferred start date to begin your application.

PhD Algebra and Number Theory - 3 Years (Full-time)

Phd algebra and number theory - 6 years (part-time), application deadlines.

We strongly recommend you submit your completed application as early as possible, particularly if you are also applying for funding or will require a visa. We may consider late applications if we have places available. All applications received by 22 January 2024 will receive full consideration for funding. Later applications will be considered until all positions are filled.

• How to apply

You must submit two references with your application.

Find out more about the general application process for postgraduate programmes:

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College of Arts and Sciences

Department of mathematics, applied mathematics, and statistics, mathematics.

A student in the traditional mathematics program must demonstrate knowledge of the basic concepts and techniques of algebra, analysis (real and complex), and topology. This includes taking all courses in the three basic areas, and successfully completing qualifying examinations in algebra and analysis.

Qualifying Examination

A doctoral student in the mathematics track must take written examinations on abstract algebra and real analysis, as well as an oral examination in his or her chosen area of specialization. Subjects include complex analysis, control and calculus of variations, differential equations, dynamical systems, functional analysis, geometry, probability, and topology.

Required Courses

A student with a master’s degree in a mathematical subject compatible with our program, as determined by the graduate committee, must take 18 credit hours of approved courses. The graduate committee will determine which of the specific course requirements stated above have been satisfied by the master’s course work.

PhD Qualifying Exams

The requirements for the PhD program in Mathematics have changed for students who enter the program starting in Autumn 2023 and later. 

Requirements for the Qualifying Exams

Students who entered the program prior to autumn 2023.

To qualify for the Ph.D. in Mathematics, students must pass two examinations: one in algebra and one in real analysis. 

Students who entered the program in Autumn 2023 or later

To qualify for the Ph.D. in Mathematics, students must choose and pass examinations in two of the following four areas: 

• real analysis
• geometry and topology
• applied mathematics

The exams each consist of two parts. Students are given three hours for each part.

Topics Covered on the Exams:

• Algebra Syllabus
• Real Analysis Syllabus
• Geometry and Topology Syllabus
• Applied Mathematics Syllabus

Check out some Past and Practice Qualifying Exams to assist your studying.

Because some students have already taken graduate courses as undergraduates, incoming graduate students are allowed to take either or both of the exams in the autumn. If they pass either or both of the exams, they thereby fulfill the requirement in those subjects. However, they are in no way penalized for failing either of the exams.

Students must pass both qualifying exams by the autumn of their second year. Ordinarily first-year students take courses in algebra and real analysis throughout the year to prepare them for the exams. The exams are then taken at the beginning of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Students who started in Autumn 2023 and later

Students must choose and pass two out of the four qualifying exams by the autumn of their second year. Students take courses in algebra, real analysis, geometry and topology, and applied math in the autumn and winter quarters of their first year to prepare them for the exams. The exams are taken during the first week of Spring Quarter. A student who does not pass one or more of the exams at that time is given a second chance in Autumn. 

Exam Schedule

Unless otherwise noted, the exams will be held each year according to the following schedule:

Autumn Quarter:  The exams are held during the week prior to the first week of the quarter. Spring Quarter:  The exams are held during the first week of the quarter.

The exams are held over two three-hour blocks. The morning block is 9:30am-12:30pm and the afternoon block is 2:00-5:00pm.

For the start date of the current or future years’ quarters please see the  Academic Calendar

Upcoming Exam Dates

Spring 2024.

The exams will be held on the following dates:

Monday, April 1st

Analysis Exam, Room 384H

Wednesday, April 3rd

Algebra Exam, Room 384I

Thursday, April 4th 

Geometry & Topology Exam, Room 384I

Friday, April 5th

Applied Math Exam, Room 384I

PhD in Mathematical Sciences

The Doctor of Philosophy (PhD) program in Mathematical Sciences represents achievement of a broad knowledge of the various branches of mathematics, of the ability to communicate mathematics in both written and oral form, and of a demonstrated creative ability in a particular branch of mathematics.

The program prepares students for careers in academia and as professional mathematicians. Students have the opportunity to work with some of the world's leading experts in a variety of research areas. The Department has particular strength in the interrelated fields of analysis , partial differential equations , and probability . Students interested in more applied directions can work with experts in  mathematical bioscience , computational finance , and  cryptography ..

Admission Requirements

Students applying for the program should have or be expecting to obtain a bachelor’s degree either in mathematics or a related field.

The following background is required:

• Multivariable calculus at the level of MATH 2063
• Ordinary differential equations at the level of MATH 2073 or MATH 2074
• Linear algebra at the level of MATH 2076
• Experience with reading and constructing mathematical proofs at the level of MATH 3001

The following background is recommended:

• Analysis at the level of MATH 3002 or MATH 5101/6001
• Abstract linear algebra at the level of MATH 5103/6003  
• Additional advanced courses in pure or applied mathematics

An official general GRE score is required for admission.  This requirement is waived for applicants with at least 6 graduate credits in a relevant field with a cumulative GPA of 3.2 or higher.  The requirement is also waived for UC undergraduates with a degree in a relevant field and a cumulative GPA of 3.5 or higher.  A quantitative score of 160 or higher is recommended.

The English proficiency requirement is met for applicants with degrees earned in English from accredited universities and colleges in  the US or other English-speaking countries .

Financial Support

Most of our PhD students receive full financial support via a teaching or research assistantship, and most are supported through their entire UC career.  Travel support is available for students to attend or present their work at conferences.  

All applicants for the PhD program are automatically reviewed for graduate assistantship eligibility at the time of application. 

• Financial aid opportunities for students in the Mathematical Sciences Department
• Tuition and fees for graduate and professional students

Application Instructions

Applicants will need to meet the minimum requirements to be considered for the program. Completed applications will be reviewed beginning February 1 . We will continue to receive applications until all positions are filled.

All application materials from international students requiring a US visa must be received prior to April 1 (but sooner is better) in order to allow time for the necessary paperwork to be processed. The visa application process can often take 90 days or more to complete.

How to apply: 

1. Create  an online application

2. Include these documents in your application:

• Three letters of recommendation. The application system will automatically send an email to each of the recommenders with a link to submit their letters. 
• Unofficial copy of transcript (official transcript will be required if you are admitted to the program). 
• GRE general test score 
• Statement of purpose/cover letter 
• English Proficiency for international students

3. Pay the application fee

UC’s CEEB college code is 1833, as established by The College Board . CEEB codes are used to ensure that test scores are sent to the correct institution. 

• More information about submitting your application materials
• FAQs for the admission process

Program Description

The credit-hour requirement includes a minimum of 90 graduate credits beyond the bachelor's degree or a minimum of 60 credits beyond a master's degree, including 7 hours in dissertation research, with a GPA of 3.3 or higher. 

All incoming PhD students are required to take the  qualifying exam before the beginning of their first semester. Students who do not pass this exam at the PhD level are placed in the appropriate 6000 - level courses. The Mathematics Qualifying Exam is based on the two-semester sequence Advanced Calculus MATH6001-6002 and the one semester course Abstract Linear Algebra MATH6003. 

All PhD students must pass four preliminary examinations . Each Preliminary Exam is offered twice a year. Examinations based on a course given during Fall Semester are offered after the end of Spring Semester and at the beginning of the following Fall Semester. Examinations based on a course given during the Spring Semester are offered at the beginning of the Fall Semester and at the beginning of the following Spring Semester.

After the preliminary examinations, an advanced examination in the area of examination of the student is required. An advanced exam may either be a written exam, a presentation or a series of presentations. The exam will be administered by a committee. Generally, this committee will form the students’ dissertation committee.

Visit the curriculum guide to learn about the required courses. More details concerning the requirements of the program are explained in the Mathematical Sciences Department’s Graduate Handbook . See the course descriptions for information on the content.

About Cincinnati

Cincinnati is a big city with a small-town feel. The cost of living is low, but the quality of life is high.  Forbes named Cincinnati the #5 most affordable city and the #9 best city for raising a family. Cincinnati has ranked  the best place to live in Ohio by U.S. News & World Report, also the fourth-best city in the country for parks . UC is home to over 10,500 graduate students, 20% of which are international students.

• Why Cincinnati  
• Estimated living expenses  (for international students)

For further information, please contact the Graduate Program Director, Dr. Robert Buckingham:

• Email: [email protected]
• Phone: 513-556-4085

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Dept. of Department of Mathematics

Earn admissions to a nationally ranked program.

Incoming Ph.D. students are expected to pass  qualifying examinations  by the end of their third semester in the Ph.D. program.

After passing the qualifying examinations, students are expected to select a thesis adviser and form a doctoral committee. The committee administers the comprehensive examination (no later than the end of the sixth semester of study) and offers counsel to the student as his research progresses.

• No credit will be given for any course in which a grade of less than B is received.
• A minimum grade point average of 3.0 is required for graduation for all advanced degrees.

Requirements

• American English Oral Communicative Proficiency Test (AEOCPT)-International students only  ( Graduate School requirement) This exam must be taken upon arrival during the week of orientation. Students who pass the exam may teach without restrictions; marginal or failing students must register for the appropriate ESL course .  
• SARI/CITI (Scholarship and Research Integrity/Collaborative Institutional Training Initiative) ( Graduate School requirement ) The  SARI@PSU program is composed of two parts: an online course, and an interactive, discussion-based component; and encompasses content that is both interdisciplinary and discipline-specific. The online portion Part 1 ), offered through the Collaborative Institutional Training Initiative (CITI), provides a common language and understanding of the history and principles of the responsible conduct of research. This requirement must be completed within the first two weeks of the semester of arrival. The discussion-based component Part 2)provides an opportunity for in-depth exploration of important issues unique to each field of study. Part 2 is completed by attendance at the Graduate Student Seminar (see below).  
• Qualifying Examinations  ( Departmental requirement ) Ph.D. students are expected to pass four qualifying examinations out of the following areas: real analysis, linear algebra, abstract algebra, complex analysis, functional analysis, topology.  All students are required to pass real analysis and complex analysis. Students may then choose between linear and abstract algebra and functional analysis and topology.  Exams must be completed by the end of their third semester in the Ph.D. program. The system of qualifying examinations is explained in detail  here:  Qualifying Exams .  
• Colloquium Attendance Requirement  ( Departmental requirement ) Students year one (1) through the semester of the Ph.D. Oral Comprehensive Examination are required to attend at least 12 colloquium talks each academic year. Four (4) of the talks may be substituted by Departmental Seminars. First-year students (only) may substitute Student Colloquium talks for some of the required Departmental Colloquium talks. Students post-comprehensive are required to attend six (6) colloquium talks with NO substitutions.  Colloquium Attendance Form for 2023-24.  
• Graduate Student Seminar Requirement ( Departmental requirement ) This is a three credit course offered every spring. It is a requirement that every student successfully completes the seminar before his or her third year of study.  
• Ph.D. Candidacy  ( Graduate School requirement ) The Department of Mathematics will recommend Ph.D. candidacy after a student has passed all qualifying examinations and successfully completed 18 credits of Mathematics courses. Admission to candidacy is conferred by the Graduate School.  
• Oral and Written English Competency  ( Graduate School and Departmental requirement ) A candidate for the degree of Doctor of Philosophy is required to demonstrate high-level competence in the use of the English language, including reading, writing, and speaking, as part of the language and communication requirements for the Ph.D. Oral competency is assessed by the GTA (Graduate Teaching Assistant) Oversight Committee. Written competency is evaluated by the students' advisor. For additional detail, please see  Oral and Written English Competency .  
• Advisers and Doctoral Committees  ( Graduate School requirement ) Consultation or arrangement of the details of the student's semester-by-semester schedule is the function of the adviser. General guidance of a doctoral candidate is the responsibility of a doctoral committee consisting of four or more active members of the Graduate Faculty, which normally includes at least two faculty in the major field and is chaired by the student's adviser. This committee is appointed by the Graduate Dean through the Office of Graduate Programs, upon recommendation of the Director of Graduate Studies. A student should have an adviser by the end of their fourth semester and must have an adviser by the end of their fifth semester in the Ph.D. program.  
• Course Requirements  ( Departmental requirement ) Students must receive a minimum grade of B in at least eleven 3-credit 500-level mathematics courses. Students must take the Graduate Student Seminar before the third year of study.  

Starting 5/8/23:  All Modes Allowed The comprehensive examination may be held fully in-person, fully remote, or hybrid with some individuals participating in-person while others participate remotely. Student preference for delivery mode should be strongly considered, but the student and adviser must agree on the mode. If the student and adviser cannot agree on the mode, the Graduate Program Head will make the final decision. Either the student or adviser can appeal the decision of the Graduate Program Head to the ( insert appropriate graduate education administration role for the unit role, e.g. Associate Dean for Graduate Education, Director of Academic Affairs, etc. ).

• Ph.D. Thesis  ( Graduate School requirement ) The ability to do independent research and competence in scholarly exposition must be demonstrated by the preparation of a thesis on some topic related to the major subject. It should represent a significant contribution to knowledge, be presented in a scholarly manner, reveal an ability on the part of the candidate to do independent research of high quality, and indicate considerable experience in using a variety of research techniques. The contents and conclusions of the thesis must be defended at the time of the final oral examination. A draft of the thesis must be submitted to the doctoral committee a month before the final oral examination.  

Starting 5/8/23:  All Modes Allowed

The final oral examination (dissertation defense) may be held fully in-person, fully remote, or hybrid with some individuals participating in-person while others participate remotely.  Student preference for delivery mode should be strongly considered, but the student and adviser must agree on the mode. If the student and adviser cannot agree on the mode, the Graduate Program Head will make the final decision. Either the student or adviser can appeal the decision of the Graduate Program Head to the ( insert appropriate graduate education administration role for the unit role, e.g. Associate Dean for Graduate Education, Director of Academic Affairs, etc. ).

• Residency Requirement  ( Graduate School requirement ) After being admitted to candidacy, the student must be a full-time graduate student as defined by the Graduate Bulletin for two consecutive semesters (excluding summers) before comprehensive examinations can be scheduled.  
• Continuous Registration  ( Graduate School requirement ) After a Ph.D. candidate has passed the comprehensive examination and has met the two-semester full-time residency requirement (above), the student must register continuously for each fall and spring semester (beginning with the first semester after both of the above requirements have been met) until the Ph.D. thesis is accepted and approved by the doctoral committee.

• Department of Mathematics >
• Graduate >

Doctoral Program (PhD)

UB's doctoral program in mathematics aims toward generating career options for our students. Additionally, the program guides students toward being prepared for research by the end of third year of coursework.

As reported by the American Mathematical Society, mathematician was the #1 rated career in CareerCast’s Job Rated 2014 report. Individual who have demonstrated a high level of mathematical acumen by obtaining a PhD in mathematics are highly prized in both the academic and private sector job markets.

The requirements below are for students admitted in Fall 2016 and later. The main steps in completing a PhD are:

(A) First Year's Coursework and Evaluation exams— Successfully completing the first year's 6 core courses and passing at least 4 out of 6 evaluation exams attached to these courses. For students interested in pursuing research in pure mathematics the 6 core courses are in algebra, analysis and geometry/topology. For students interested in pursuing research in applied mathematics the 6 core courses are in analysis, numerical analysis and methods in applied mathematics.

(B) Oral Examination and Advancing to Candidacy— An oral examination covering material in advanced topics and research ideas in the student's chosen area of research. This oral examination is also the final requirement for advancement to candidacy and should be taken before the end of the student's third year.

(C) PhD Thesis and Final Oral Examination— Writing a dissertation and passing an oral defense.The dissertation must consist of original research of sufficient quality for publishing in a respectable mathematics journal.

After the 1st year's course work, the student will take more advanced courses at the 600 level and 700/800 level topics course. Entering their 3rd year, students will focus on their preferred area of research. Advancement to candidacy and dissertation work requires passing an oral exam. Students should pass their oral examination prior to the end of the 3rd year of the program.

In addition to these primary steps, the program offers a 1st year mentoring seminar meant to help students in their career development and management. Topics covered include: study mathematics; using LaTex; media in research mathematics; documenting your achievements; writing, editing and publishing mathematics; seminars, conferences and workshops; and, job options for PhD's in mathematics.  

This mentoring seminar will also include faculty talks directed at graduate students, presenting their area of research.

Both the MA and the PhD degrees have residency requirements: one year for the MA and two years for the PhD.

On this page

Phd program requirements.

The main steps in obtaining a PhD in mathematics are: (A) Satisfactory completion of first year's coursework and evaluation exams; (B) Passing oral exam in intended area of research and advancing to candidacy; and, (C) Writing the dissertation and successfully defending it in a final oral exam. The aspects of each step are more fully discussed on this page.

(A) First Year's Coursework and Evaluation exams

The course schedule outlined below is for students in the PhD program who are supported by a teaching assistantship and tuition fellowship. It is 9-credits per semester. For students who do not have support an additional 3-credit course is require so as to be a full time student. 

Learning mathematics is a shared enterprise. Thus, all members of an entering doctoral class advance through the first year coursework as a cohort.

Fall semester:

• MTH 534, Basic Measure Theory.
• MTH 519, Introduction to Abstract Algebra.
• MTH 527, Introduction to Topology I.  
• MTH 539, Methods of Applied Mathematics.
• MTH 537, Introduction to Numerical Analysis I.
• One of:  MTH 534, Basic Measure Theory; or MTH 519, Introduction to Abstract Algebra; or MTH 527, Introduction to Topology I.

Spring semester offering:

• MTH 625, Complex Variables.
• MTH 520, Advanced Linear Algebra.
• MTH 528, Introduction to Topology II.  
• MTH 540, Methods of Applied Mathematics II.
• MTH 538, Introduction to Numerical Analysis II.
• One of:  MTH 639 Fourier Analysis; or, MTH 625, Complex Variables. MTH 520, Advanced Linear Algebra. MTH 528, Introduction to Topology II.

Evaluation Exams: Attached to each first year course is an evaluation exam. This exam will be given during the regularly scheduled final exam time. All first year evaluation exams are pass/fail.  To continue in the PhD program a student needs to achieve at least 4-out-of-6 exam passes. To continue in the MA program a student needs to achieve at least a 3-out-of-6 exam passes. To be in good standing in any graduate program a student needs a GPA of B or above.

Deficiency:  Students whose performance at the end of their 1st year is judged to be significantly insufficient by the Graduate Director will be dismissed from the program before the beginning of their 2nd year. Students who are marginally below the mark (e.g., pass 2 out of 4 exams or better at PhD level) and/or are marginally below the required B-GPA level, so that they can still advance with their original cohort, have an opportunity to retake the relevant exams in the final exam week of the Fall and Spring semesters in their 2 nd  year. If the student passes these “make ups’’ (i.e., pass 4-out-of-6 in total for PhD and 3-out-of-6 for MA), then the student will be allowed to advance through the program along with their original cohort. If not, then the student will be dismissed from the program.

(B) Oral Examination and Advancing to Candidacy

After the first year's course work, the student will take more advanced courses at the 600 level and 700/800 level topics course. Students also typically arrange individual reading courses with professors and participate in area seminars.

Entering their third year, students will be focusing on their preferred area of research and the faculty with whom they would like to work. Students will be required to form an oral examination committee of two or three faculty members chaired by a potential thesis advisor.

Students will work with their committee to prepare a syllabus outlining topics to be covered in the oral examination including a bibliography of books and/or articles. Typically the topics to be covered in the oral examination should be at the level of 600 to 800 level courses and include material that the student learned individually.

The syllabus must be approved by the Graduate Director’s office and the student’s committee members. Students should pass their oral examination prior to the end of the third year of the program.

(C) PhD Thesis and Final Oral Examination

The final departmental steps in attaining the degree is completion of a dissertation that must consist of original research of sufficient quality for publishing in a respectable mathematics journal. It is not unusual for the mathematics in a single dissertation to generate two or three published manuscripts.

PhD Thesis Template

Student resources and related links.

Jenny Russell

Assistant to the Graduate Director

Department of Mathematics

227 Mathematics Building, Buffalo, NY 14260-2900

Phone: 716-645-8782; Fax: 716-645-5039

Email: [email protected]

For those students admitted to the program in 2015, the prior requirements remain in effect.

The main steps in completing a PhD are: passing qualifying examinations; and, writing a dissertation.

The qualifying examinations are taken in several parts. During the first year of full-time study, the student must pass the First Qualifying Examination, an exam on basic material from undergraduate algebra and analysis. During the second year, the student must pass a more advanced, but quite flexible Second Qualifying Examination based on courses at the 600 level and above. By the end of the third year, the student must pass another exam, the nature of which will vary from student to student, and depends primarily on the student's area of study and thesis advisor.

The dissertation must consist of original research of sufficient quality to be published in a respectable mathematics journal. Upon completion of the second qualifying exam, the student will choose (in consultation with the director of graduate studies) a doctoral committee, the chair of which will direct the thesis research. Upon completion of the thesis, the student must pass a final oral examination administered by the department.

The week before classes begin in August, all new M.A. and Ph.D. students must take the First Qualifying Examination . The syllabus for this exam is based on undergraduate analysis and algebra (including linear algebra). This exam is given before classes begin to enable the student and the director of graduate studies to refer to its results while deciding the most appropriate courses for the student.

The main steps in obtaining a PhD are passing the qualifying examinations, writing a thesis, and passing a final oral examination on this thesis. The departmental regulations concerning each of these are given below. The regulations are interpreted by the graduate studies committee which, on written petition from a student, may permit deviations from the rules, provided there are exceptional circumstances. In addition to the departmental regulations, there are university requirements which must also be satisfied.

Admission with Advanced Standing At the time of admission to UB's Graduate School, the director of graduate studies may decide that certain students have advanced standing of one or two semesters of graduate work, depending on UB Graduate School requirements. This will be done after examining the graduate records of the students and taking account of his previous courses, the institutions where he studied, his proficiency in English (TOEFL), etc. It will be clear from what follows that such students will have to fulfill various requirements more quickly than normally admitted students.

Definition of Total Semesters of Graduate Work The sum of the semesters of graduate work as defined by (i) and (ii) below yields the total semesters of graduate work which will simply be called "semesters of graduate work".

(i) A student admitted with graduate coursework may credited with one or two semesters of graduate work, according to Graduate School requirements.

(ii) For every semester at SUNYAB that a student is registered for fewer than nine credit hours, the credit hours are to be totaled and divided by nine. The result, rounded down to the next integer, will also be counted as semesters of graduate work. In no event will a student be said to have completed more than two semesters of academic work in one calendar year.

Deficiency A student is considered to have a deficiency if in the first semester as a graduate student at UB,  the student officially enrolls in, and completes, Math 519 (introductory algebra) or Math 531 (introductory real variables). The student should base her/his decision on whether to take these courses on advice from the director of graduate studies and on evaluation of the student's knowledge in algebra and analysis by the relevant area committees.

First Qualifying Examination

The First Qualifying Exam is a three-and-a-half-hour written examination based on a syllabus covering introductory real variables at the level of MTH 431-432, introductory abstract algebra at about the level of MTH 419, and linear algebra at about the level of MTH 420. The examination is given twice a year, during the week prior to the beginning of each semester.

The purpose of the first examination is to assist the director of graduate studies and the student in deciding soon after the student's entry into the UB Graduate School, whether or not the student will be admitted to the the PhD program in mathematics.

Normally, to remain in the PhD program, a student is required to pass this examination within the first two years of graduate work. A student who entered with a deficiency is not required to pass this examination until the first opportunity after completiing two semesters of graduate work. See the Syllabus for the First Qualifying Examination (Revised 04/25/13) attached as a pdf, below.

Second Qualifying Exam This consists of two three-hour area examinations, selected by each student from the following four choices: ALGEBRA; ANALYSIS; GEOMETRY/TOPOLOGY; and DIFFERENTIAL EQUATIONS. It is the purpose of the second qualifying examination to insure that each student has a rudimentary command of at least two "core" areas of mathematics.

To remain in the PhD program a student is required to obtain a grade of A or B for one of the area examinations no later than the beginning of his fourth semester of graduate work and an average of at least B for both of the area exams no later than the beginning of his fifth semester. Students may repeat the examinations, within the time limit, without penalty and are encouraged to take at least one of the examinations as early as possible. See Information on the Second Quaifying Examination, attached as a pdf, below.

Doctoral Committee During the semester in which he completes the Second Qualifying Examination, each student will select a major professor, who is a member of the graduate faculty, in consultation with the director of graduate studies. The latter and the major professor will then choose the student's doctoral committee, consisting of at least three members of the faculty with the major professor as chair.

Admission to Candidacy The student's doctoral committee will set the requirements for admission to candidacy. These are subject to the approval of the director of graduate studies and may include, but are not restricted to, any of the following: an oral examination on "research level" material, a project, a series of lectures on "research level" mathematics, or a written qualifying examination in another department. These requirements must be satisfied by the end of the sixth semester of graduate work.

Language Requirements There are no language requirements.

Additional Course Work Before the final oral exam, each student should pass, with a grade of A, B , or S , two one-semester graduate course in subjects other than those of his or her second qualifying exam. These courses are to be approved by the director of graduate studies. Each PhD student must complete 72-credit hours from: (a) selected 500 level mathematics courses; (b) 600-800 level Mathematics courses, with the exception of thesis guidance, seminar courses, and other courses of this nature; (c) courses designated by his/her major professor.

PhD Thesis and Final Oral Examination

The final departmental steps in attaining the degree of Doctor of Philosophy are:

1. Completion of a thesis satisfactory to the major professor and the student's doctoral committee;

2. Approval by the UB Graduate School that the student proceed to examination on his/her thesis at a final oral examination;

3. Submission of the thesis to each member of the doctoral committee at least three weeks prior to the final oral examination;

4. Passing the final oral examination.

Program Requirements for students admitted Fall 2015 and earlier

Mathematics, PhD

• Program description
• At a glance
• Degree requirements
• Admission requirements
• Tuition information
• Application deadlines
• Program learning outcomes
• Career opportunities
• Contact information

Algebra, Equations, Geometry, Mathematics, analysis, approved for STEM-OPT extension

Are you interested in understanding the true depth of knowledge in the intradisciplinary subfields within mathematics? Discover important connections between different areas of mathematics and their applications using studies in algebra, topology, geometry, probability, analysis and logic.

The PhD program in mathematics is intended for students with exceptional mathematical ability. The program emphasizes a solid mathematical foundation and promotes innovative scholarship in mathematics and its many related disciplines.

The School of Mathematical and Statistical Sciences has very active research groups in analysis, number theory, geometry and discrete mathematics.

This program may be eligible for an Optional Practical Training extension for up to 36 months. This OPT work authorization term may help international students gain skills and experience in the U.S. Those interested in an OPT extension should review ASU degrees that qualify for the STEM-OPT extension at ASU's International Students and Scholars Center website.

The OPT extension only applies to students on an F-1 visa and does not apply to students completing the degree through ASU Online.

• College/school: The College of Liberal Arts and Sciences
• Location: Tempe

84 credit hours, a written comprehensive exam, a prospectus and a dissertation

Required Core (3 credit hours) MAT 501 Geometry and Topology of Manifolds I (3) or MAT 516 Graph Theory I (3) or MAT 543 Abstract Algebra I (3) or MAT 570 Real Analysis I (3)

Other Requirements (3 credit hours) MAT 591 Seminar (3)

Electives (24-39 credit hours)

Research (27-42 credit hours) MAT 792 Research

Culminating Experience (12 credit hours) MAT 799 Dissertation (12)

Additional Curriculum Information Electives are to be chosen from math or related area courses approved by the student's supervisory committee.

Students must pass:

• two qualifying examinations
• a written comprehensive examination
• an oral dissertation prospectus defense

Students should see the department website for examination information.

Each student must write a dissertation and defend it orally in front of five dissertation committee members.

Applicants must fulfill the requirements of both the Graduate College and The College of Liberal Arts and Sciences.

Applicants are eligible to apply to the program if they have earned a bachelor's or master's degree in mathematics or a closely related area from a regionally accredited institution.

Applicants must have a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in the last 60 hours of their first bachelor's degree program or a minimum cumulative GPA of 3.00 (scale is 4.00 = "A") in an applicable master's degree program.

All applicants must submit:

• graduate admission application and application fee
• official transcripts
• statement of education and career goals
• three letters of recommendation
• proof of English proficiency

Additional Application Information An applicant whose native language is not English must provide proof of English proficiency regardless of their current residency.

Additional eligibility requirements include competitiveness in an applicant pool as evidenced by coursework in linear algebra (equivalent to ASU course MAT 342 or MAT 343) and advanced calculus (equivalent to ASU course MAT 371), and it is desirable that applicants have scientific programming skills.

Program learning outcomes identify what a student will learn or be able to do upon completion of their program. This program has the following program outcomes:

• Apply advanced mathematical skills in coursework and research.
• Address an original research question in mathematics.
• Able to complete original research in theoretical mathematics.

Graduates of the doctoral program in mathematics possess sophisticated mathematical skills required for careers in many different sectors, including education, industry and government. Potential career opportunities include:

• faculty-track academic
• finance and investment analyst
• mathematician
• mathematics professor, instructor or researcher
• operations research analyst
• statistician

School of Mathematical and Statistical Sciences | WXLR A213 [email protected] 480-965-3951

Abstract Algebra: Theory and Applications

(4 reviews)

Thomas W. Judson, Stephen F. Austin State University

Copyright Year: 2016

ISBN 13: 9781944325022

Publisher: University of Puget Sound

Language: English

Formats Available

Conditions of use.

Free Documentation License (GNU) Free Documentation License (GNU)

Learn more about reviews.

Reviewed by Malik Barrett, Assistant Professor, Earlham College on 6/24/19

Judson covers all of the basics one expects to see in an undergraduate algebra sequence. That is, some review from discrete math/intro to proofs (chapters 1-2), and elementary group theory including chapters on matrix groups, group structure,... read more

Comprehensiveness rating: 5 see less

Judson covers all of the basics one expects to see in an undergraduate algebra sequence. That is, some review from discrete math/intro to proofs (chapters 1-2), and elementary group theory including chapters on matrix groups, group structure, actions, and Sylow theorems.

The coverage of ring theory is slimmer, but still relatively "complete" for a semester of undergraduate study. Three chapters on rings, one on lattices, a chapter reviewing linear algebra, and three chapters on field theory with an eye towards three classical applications of Galois theory. I will note here that Judson avoids generators and relations.

The coverage is all fairly standard, with excepting the definition of Galois group (see accuracy), and the referencing system in the HTML version is extremely convenient. For example, Judson leverages HTML so that proofs are collapsed (but can be expanded) which allows him to clean up the presentation of each section and include full proofs of earlier results when useful as references. The index uses a similar approach, choosing to display a collapsed link to the first paragraph in which the term is used, which is often a formal definition. There are no pages displayed, but there is a google search bar to scan the book with. Given the searchability, the index style is an interesting choice.

Since Judson includes _a lot_ of Sage which he uses to expand, clarify, or apply theory from the text, a fairly standard presentation of the theory, and includes hints/solutions to selected exercises, the textbook is very comprehensive.

Content Accuracy rating: 4

I've noticed very few outright errors in the text proper. However, of primary note is Judson's non-standard (in my experience) definition of Galois group as the automorphism group Aut(E/F) of an arbitrary field extension E/F. He defines this before he's defined fixed fields (ala Artin), or normal/separable extensions. All of the exercises use this definition as well, and so I chose to (mostly) avoid the chapter on Galois theory in favor of a more standard presentation.

There _are_ some errors in the exercises, however, like the inclusion of unnecessary or irrelevant parts, or typos. But I came across very few of these in my problem sets.

Relevance/Longevity rating: 5

Modern applications are sprinkled throughout the text that informs the students of the value of the material beyond theoretical. Judson does this in practical ways given that Sage is such a big component of the book, and so there are many exercises and descriptions that stress this relevance.

Clarity rating: 4

Judson's writing is direct and effective. I find his style clean and easy to follow. However, there are instances where there are big jumps between what some beginning exercises assume and what was presented explicitly in the chapter which confused many of my students. For instance, there is a dearth of examples of how to compute minimal polynomials and extension degrees (and the subtleties involved), and so the instructor has to provide the strategies necessary to solve parts of the first two problems.

Consistency rating: 5

The book is consistent in language, tone, and style. The only inconsistencies I've noticed involve the occasional definition appearing inline (usually in a sentence motivating the definition) instead of set aside in a text box. Defined terms _are_ still shown in bold, though. Still, it can make it hard to locate the precise definition quickly by scanning the section, but happens so rarely I won't detract a point.

Modularity rating: 5

Judson is very direct, and so his chapters are very focused. Moreover, many sections are punctuated, perhaps including no more than several definitions and propositions along with a historical note. So it's quite easy to divide the material into tight, bite-sized portions along the sections of the book, with a few exceptions, i.e., sections that run -much- longer and denser than average, like the section on field automorphisms.

Many sections and some chapters are written in a way that relies minimally on previous material which allows one to omit them or change the order of presentation without too much fuss. For instance, it's easy to cover the material on matrix groups and symmetry (chapter 12) right after the intro coverage of groups (chapter 3) if you want more concrete examples. Or omit the chapters on integral domains (with some minimal adjustment), lattices, and linear algebra if one is making a push to fields and Galois theory.

Organization/Structure/Flow rating: 5

The text has a relatively linear progression, with some exceptions. The exceptions aren't detractions, though, and allow for modularity or digressions to applications.

Interface rating: 5

The UI of the text is amazingly clean and efficient. Google search makes scanning the book quick and easy, the collapsible table of contents and the sidebar makes jumping around in the text simple. Sage can be run on the page itself making the Sage section quite effective. One can even right-click on rendered LaTeX, like tables, and copy the underlying code (which is super convenient for Cayley tables).

Grammatical Errors rating: 5

I recall no major grammatical errors.

Cultural Relevance rating: 5

Judson sticks to the math, so the text is pretty impersonal. Even the historical notes are fact-based accounts.

I used the book for a year-long algebra sequence and was fairly happy with the outcome. Beyond the first two sections of the Galois theory chapter being too non-standard for my tastes, I had few complaints and will very likely use the text again. The problem bank is also very good and they generally complement the material from the chapters quite well.

Reviewed by Andrew Misseldine, Assistant Professor, Southern Utah University on 6/19/18

This textbook is recommended for a upper division undergraduate course on abstract algebra and contains enough materials to cover a two-semester sequence, with particular emphasis placed on groups, rings, and fields. The group theory contains... read more

This textbook is recommended for a upper division undergraduate course on abstract algebra and contains enough materials to cover a two-semester sequence, with particular emphasis placed on groups, rings, and fields. The group theory contains all the main topics of undergraduate algebra, including subgroups, cosets, normal subgroups, quotient groups, homomorphisms, and isomorphism theorems and introduces students to the important families of groups, with a particular emphasis on finite groups, such as cyclic, abelian, dihedral, permutation, and matrix groups. The textbook also includes more advanced topics such as structure of finite abelian groups, solvable groups, group actions, and Sylow Theory. The coverage of rings is equally comprehensive including the important topics of ideals, domains, fields, homomorphisms, polynomials, factorization, field extensions, and Galois Theory. The book is accompanied with a comprehensive index of topics and notation as well of solutions to selected exercises.

Content Accuracy rating: 5

The content of the textbook is very accurate, mathematically sound, and there are only a few errors throughout. The few errors which still exist can be reported to the author via email who appears to be very welcoming to suggestions or corrections from others. The author updates the textbook annually with corrections and additions.

This textbook follows the classical approach to teaching groups, rings, and fields to undergraduate and will retain its value throughout the years as the theory and examples will not be changing. It is possible that some of the applications included, mostly related to computer science, could eventually become obsolete as new techniques are discovered, but this will probably not be too consequential to this text which is a math book and not a compute science textbook. The applications of algebra can still be interesting and motivating to the reader even if they are not the state-of-the-art. The author updates the textbook annually with corrections and is very welcoming to suggestions or corrections from others.

Overall, the textbook is very clear to read for those readers with the appropriate background of set theory, logic, and linear algebra. Proofs are particularly easy to follow and are well-written. The only real struggle here is in the homework exercises. Occasionally, the assumptions of the homework are not explicit which can lead to confusion for the student. This is often the fault that the exercises are collected for the entire chapter and not for individual sections. It can sometimes be a chore for instructors to assign regular homework because they might unintentionally assign an exercise which only involves vocabulary from an early section but whose proofs required theory from later in the chapter.

The author is consistent in his approach to both the theory and applications of abstract algebra, which matches in style many available textbooks on abstract algebra. In particular, the book's definitions and names of important theorems are in harmony with the greater body of algebraists. It is also consistent with its notation, although sometimes this notations deviates from the more popular notations and often fails to mention alternative notations used by others. A comprehensive notation index is included with references to the original introduction of the notation in the text. Regrettably, no similar glossary of terms exists except the index, which is should be sufficient for most readers.

Modularity rating: 4

The textbook is divided into chapters, sections, and subsections, with exercises and supplementary materials placed in the back of each chapter or at the end of the book. These headings and subheadings lead themselves naturally to how an instructor might parse the course material into regular lectures, but, dependent of the amount of detail desired by the instructor, these subsections do not often produce 50-minute lectures. The textbook's preface includes a dependency chart to help an instructor decide on the order of topics if time restricts complete coverage of the topics. The textbook could be easily adapted for a two semester sequence with the first semester covering groups and the second covering rings and fields or a single semester course which introduces both groups and rings while skipping the more advanced topics. The application chapters/sections can easily be included into the course or omitted from the course based upon the instructor's interest and background with virtually no interruption to the students. Some chapters include a section of "Additional Exercises" which include exercises about topic not covered in the textbook but adjacent to the topics introduced. Although these sections are prefaced by some explanation of the exploratory topic, rarely are these topics thorough explained which might leave student grossly confused and require the instructor to supplement the textbook on any exercises assigned from here.

All sections follow the basic template of first introducing new definitions followed by examples, theorems, and proofs (although counterexamples are included, the presentation could benefit from additional counterexamples) and further definitions, examples, and theory are introduced as appropriate. Each chapter is concluded with a historical note, exercises for students, and references and suggested readings. Additionally, each chapter includes a section about programming in Sage relevant to the chapter contents with accompanying exercise, but this section is only available in the online version, not the downloadable or print versions. The first chapters review prerequisite materials including set theory and integers, which can be skipped by those students with a sufficient background without any loss. This book takes a "group-first" approach to introductory abstract algebra with rings, fields, vector spaces, and Boolean algebras introduced later. Throughout the textbook, in addition to the examples and theory, there are several practical applications of abstract algebra with a particular emphasis on computer science, such as cryptography and coding theory. These application sections/chapters can be easily included into the course without much extra preparation for the instructor or omitted at no real disruption to the student.

This textbook was authored using PreTeXt, which designed for typesetting mathematical documents and allow them to be converted into multiple formats. This textbook is available in an online, downloadable pdf, and print version. All three versions have solid format, especially in regard to the mathematical typesetting and graphics. The online version is available in both English and Spanish, where the interface and readability are equally of high quality.

The textbook appears to be absent of regular grammatical or mathematical errors, although a few might be present. The few errors which still might exist can be reported to the author via email who appears to be very welcoming to suggestions or corrections from others. The author updates the textbook annually with corrections and additions. For the purposes of this review, the English version of the textbook was reviewed. The reviewed makes no claim about the quality of the grammar of the Spanish version which was translated by Antonio Behn from the author's original English version.

Culture is not really a concern for theoretical mathematics textbooks which focuses almost entire on mathematical content knowledge and theory and not so much on people or their relationships. The textbook is devoid of culturally insensitive of offensive materials. Many chapters end with historical notes about mathematicians who helped to develop the chapter's materials. These notes typically follow the traditional Western European narrative of abstract algebra's development and is fairly homogeneous. Efforts could be made to include a more diverse and international history of algebra beyond Europe. For example, there is no historical note about the Chinese Remainder Theorem other than a sentence to explain why its name includes the word "Chinese." The textbook, originally written in English, now includes a complete Spanish edition, which is a massive effort for any textbook to be more inclusive.

This has been one of my absolute favorite textbooks for teaching abstract algebra. In fact, I think Judson's book is a golden standard for what a high-quality, mathematical OER textbook should be. It has created using the very impressive PreTeXt. In addition to the different formats, this book includes SAGE exercises. It has enough material to fill the usual two-semester course in undergraduate abstract algebra.

Reviewed by Nicolae Anghel, Associate Professor, University of North Texas on 4/11/17

This is a two-in-one book: a theoretical part and a computational part. Initially the OTL contained a 2014 version of the book, which only made tangential reference to the SAGE computational system. I downloaded from the author’s website the full,... read more

This is a two-in-one book: a theoretical part and a computational part. Initially the OTL contained a 2014 version of the book, which only made tangential reference to the SAGE computational system. I downloaded from the author’s website the full, 2016 version, which eventually was also made into the OTL default. The theoretical part of the book is certainly adequately comprehensive, covering evenly the proposed material, and being supported by judiciously chosen exercises. The computational part also seems to me comprehensive enough, however one should not take my word for it as this side exceeds my areas of expertise and interest.

The parts that I checked, at random, were very accurate, so I have no reason to believe that the book was not entirely accurate. However, only after testing the book in the classroom, which I intend to do soon, can I certify this aspect.

The material is highly relevant for any serious discussion on math curriculum, and will live as long as mankind does.

Clarity rating: 5

For me as instructor the book was very clear, however keep in mind that this was not the first source for learning the material. Things may be different for a beginning student, who sees the material for the first time. Again, a judgment on this should be postponed until testing the book in the classroom.

The book is consistent throughout, all the topics being covered thoroughly and meaningfully.

I have no substantive comments on this topic.

The book, maybe a little too long for its own good, is divided into 23 chapters. The flow is natural, and builds on itself. The structure of each chapter is the same: After adequately presenting the material (conceptual definitions, theorems, examples), it proceeds to exercises, sometimes historical notes, references and further readings, to conclude with a substantial computational (based on SAGE syntax) discussion of the material, also including SAGE exercises. The applications to cryptography and coding theory highlight the practical importance of the material. I particularly liked the selection of exercises.

Another big advantage of a free book is that the student does not have to print all of it, certainly not all of it at the same time. This is a big plus, since with commercial books most of the time a student buys a book and only a fraction of it is needed in a course.

Written in a conversational, informal style the book is by and large free of grammatical errors. There are about a dozen minor mistakes, such as concatenated words or repeated words.

The historical vignettes are sweet. Maybe adding pictures of the mathematicians involved would not be a bad thing.

I liked the book, but I like more the concept of free access to theoretical and practical knowledge. Best things in life should essentially be free: air, water, …, education. I will make an effort to use open textbooks whenever possible.

Reviewed by Daniel Hernández, Assistant Professor, University of Kansas on 8/21/16

This book is introductory, and covers the basic of groups, rings, fields, and vector spaces. In addition, it also includes material on some interesting applications (e.g., public key cryptography). In terms of covering a lot of topics, the book... read more

This book is introductory, and covers the basic of groups, rings, fields, and vector spaces. In addition, it also includes material on some interesting applications (e.g., public key cryptography). In terms of covering a lot of topics, the book is certainly comprehensive, and contains enough material for at least a year-long course for undergraduate math majors. A "dependency chart" in the preface should be very useful when deciding on what path to take through the text.

One noteworthy feature of this book is that it incorporates the open-source algebra program Sage. While the .pdf copy I found through the OTN website only included a not-very-serious discussion of Sage at the end of most exercise sets, the online textbook found at

http://abstract.pugetsound.edu/aata/

appears to contain a much more substantial discussion of how to use Sage to explore the ideas in this book. I admit that I didn't explore this feature very much.

Though I have not checked every detail (the book is quite long!), there do not appear to be any major errors.

The topics covered here are basic, and will therefore not require any real updates.

The book is also written in such a way that it should be easy to include new sections of applications.

I would say that this this book is well-written. The style is somewhat informal, and there are plenty of illustrative examples throughout the text. The first chapter also contains a brief discussion of what it means to write and read a mathematical proof, and gives many useful suggestions for beginners.

Through I didn't read every proof, in the ones I did look at, the arguments convey the key ideas without saying too much. The author also maintains the good habit of explicitly recalling what has been proved, and pointing out what remains to be done. In my experience, it is this sort of mid-proof "recap" is helpful for beginners.

The terminology in this text is standard, and appears to be consistent.

Each chapter is broken up into subsections, which makes it easy to for students to read, and for instructors to assign reading. In addition, this book covers modular arithmetic, which makes it even more "modular" in my opinion!

Organization/Structure/Flow rating: 4

It seems like there is no standard way to present this material. While the author's choices are perfectly fine, my personal bias would have been to discuss polynomial rings and fields earlier in the text.

The link on page v to

abstract.pugetsound.edu

appears to be broken.

My browser also had some issues when browsing the Sage-related material on the online version of this text, but this may be a personal problem.

I did not notice any major grammatical errors.

I'm not certain that this question is appropriate for a math textbook. On the other hand, I'll take this as an opportunity to note that the historical notes that appear throughout are a nice touch.

The problem sets appear to be substantial and appropriate for a strong undergraduate student. Also, many sections contain problems that are meant to be solved by writing a computer program, which might be of interest for students studying computer science.

I am also slightly concerned that the book is so long that students may find it overwhelming and hard to sift through.

Table of Contents

• Preliminaries
• The Integers
• Cyclic Groups
• Permutation Groups
• Cosets and Lagrange's Theorem
• Introduction to Cryptography
• Algebraic Coding Theory
• Isomorphisms
• Normal Subgroups and Factor Groups
• Homomorphisms
• Matrix Groups and Symmetry
• The Structure of Groups
• Group Actions
• The Sylow Theorems
• Polynomials
• Integral Domains
• Lattices and Boolean Algebras
• Vector Spaces
• Finite Fields
• Galois Theory

Ancillary Material

About the book.

This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics. Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly.

Until recently most abstract algebra texts included few if any applications. However, one of the major problems in teaching an abstract algebra course is that for many students it is their first encounter with an environment that requires them to do rigorous proofs. Such students often find it hard to see the use of learning to prove theorems and propositions; applied examples help the instructor provide motivation.

This text contains more material than can possibly be covered in a single semester. Certainly there is adequate material for a two-semester course, and perhaps more; however, for a one-semester course it would be quite easy to omit selected chapters and still have a useful text. The order of presentation of topics is standard: groups, then rings, and finally fields. Emphasis can be placed either on theory or on applications. A typical one-semester course might cover groups and rings while briefly touching on field theory, using Chapters 1 through 6, 9, 10, 11, 13 (the first part), 16, 17, 18 (the first part), 20, and 21. Parts of these chapters could be deleted and applications substituted according to the interests of the students and the instructor. A two-semester course emphasizing theory might cover Chapters 1 through 6, 9, 10, 11, 13 through 18, 20, 21, 22 (the first part), and 23. On the other hand, if applications are to be emphasized, the course might cover Chapters 1 through 14, and 16 through 22. In an applied course, some of the more theoretical results could be assumed or omitted. A chapter dependency chart appears below. (A broken line indicates a partial dependency.)

About the Contributors

Thomas W. Judson,  Associate Professor, Department of Mathematics and Statistics, Stephen F. Austin State University. PhD University of Oregon.

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Basic Abstract Algebra

An introduction to abstract algebra, brief talk on ideological and political teaching in abstract algebra, how mathematicians assign homework problems in abstract algebra courses, an invitation to abstract algebra, verifying non-isomorphism of groups.

The concept of isomorphism is central to group theory, indeed to all of abstract algebra. Two groups {G, *} and {H, ο}are said to be isomorphic to each other if there exists a set bijection α from G onto H, such that $$\left( {a\;*\;b} \right)\alpha = \left( a \right)\alpha \; \circ \;(b)\alpha $$ for all a, b ∈ G. This can be illustrated by what is usually known as a commutative diagram:

On the Exponential Diophantine Equation (132m) + (6r + 1)n = z2

Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation  (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers. Results of the present paper show that the exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers, has no solution in the whole number.

Abstract Algebra

Ring hypothesis is one of the pieces of the theoretical polynomial math that has been thoroughly used in pictures. Nevertheless, ring hypothesis has not been associated with picture division. In this paper, we propose another rundown of similarity among pictures using rings and the entropy work. This new record was associated as another stopping standard to the Mean Shift Iterative Calculation with the goal to accomplish a predominant division. An examination on the execution of the calculation with this new ending standard is finished. In spite of the fact that ring hypothesis and class hypothesis from the start sought after assorted direction it turned out during the 1970s – that the investigation of functor groupings furthermore reveals new plots for module hypothesis.

(m, n)-Ideals in Semigoups Based on Int-Soft Sets

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, and topological spaces. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader framework of fuzzy setting. In this paper, we introduce the notions of int-soft m , n -ideals, int-soft m , 0 -ideals, and int-soft 0 , n -ideals of semigroups by generalizing the concept of int-soft bi-ideals, int-soft right ideals, and int-soft left ideals in semigroups. In addition, some of the properties of int-soft m , n -ideal, int-soft m , 0 -ideal, and int-soft 0 , n -ideal are studied. Also, characterizations of various types of semigroups such as m , n -regular semigroups, m , 0 -regular semigroups, and 0 , n -regular semigroups in terms of their int-soft m , n -ideals, int-soft m , 0 -ideals, and int-soft 0 , n -ideals are provided.

A transition to abstract algebra

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Including number theory, algebraic geometry, and combinatorics

We have large groups of researchers active in number theory and algebraic geometry, as well as many individuals who work in other areas of algebra: groups, noncommutative rings, Lie algebras and Lie super-algebras, representation theory, combinatorics, game theory, and coding.

Chairs: George Bergman and Tony Feng

Algebra Faculty, Courses, Dissertations

Senate faculty, graduate students, visiting faculty, meet our faculty, george m. bergman, richard e. borcherds, sylvie corteel, david eisenbud, edward frenkel, vadim gorin, mark d. haiman, robin c. hartshorne, tsit-yuen lam (林節玄), hannah k. larson, hendrik w. lenstra, jr., ralph mckenzie, david nadler, andrew p. ogg, arthur e. ogus, martin olsson, alexander paulin, nicolai reshetikhin, john l. rhodes, kenneth a. ribet, marc a. rieffel, thomas scanlon.

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At a Crossroads—Reconsidering the Goals of Autism Early Behavioral Intervention From a Neurodiversity Perspective

• 1 Department of Psychiatry and Behavioral Sciences, Duke University, Durham, North Carolina
• 2 Duke Center for Autism and Brain Development, Duke University, Durham, North Carolina
• 3 Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina
• Comment & Response Neurodiversity and Early Autism Sarah Bernard, MBBS; Bernadette Grosjean, MD; Laura Caravallah, MD JAMA Pediatrics
• Comment & Response Neurodiversity and Early Autism Inge-Marie Eigsti, PhD; Teresa Girolamo, PhD; Deborah Fein, PhD JAMA Pediatrics

The neurodiversity perspective posits that each person has a unique brain and a unique combination of traits and abilities and asserts that many challenges faced by autistic individuals stem from a lack of fit between the characteristics of autistic people and society’s expectations and biases. The neurodiversity movement is akin to a civil rights movement. Among its goals are reducing stigma, increasing accessibility, and ensuring that autistic individuals’ voices are represented in decisions about autism research, policy, and clinical practice. The neurodiversity movement is having a growing influence on the scientific community, as evidenced in the recent pause in a large autism genetic study based on concerns raised by the autism community. 1 It is also affecting autism practitioners as, increasingly, parents are expressing reservations about enrolling their child in early intervention programs, citing concerns that such programs do not value neurodiversity and, instead, prioritize changing their child’s behavior to fit neurotypical norms.

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Dawson G , Franz L , Brandsen S. At a Crossroads—Reconsidering the Goals of Autism Early Behavioral Intervention From a Neurodiversity Perspective. JAMA Pediatr. 2022;176(9):839–840. doi:10.1001/jamapediatrics.2022.2299

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