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Abstract algebra is a broad field of mathematics, concerned with algebraic structures such as groups , rings , vector spaces , and algebras . On the 12-hour clock, \(9+4=1\), rather than 13 as in usual arithmetic

Roughly speaking, abstract algebra is the study of what happens when certain properties of number systems are abstracted out; for instance, altering the definitions of the basic arithmetic operations result in a structure known as a ring, so long as the operations are consistent.

For example, the 12-hour clock is an example of such an object, where the arithmetic operations are redefined to use modular arithmetic (with modulus 12). An even further level of abstraction--where only one operation is considered--allows the clock to be understood as a group. In either case, the abstraction is useful because many properties can be understood without needing to consider the specific structure at hand, which is especially important when considering the relationship(s) between structures; the concept of a group isomorphism is an example.

Levels of Abstraction in Abstract Algebra

Group theory.

• Ring Theory

Other Applications of Abstract Algebra

It is possible to abstract away practically all of the properties found in the "usual" number systems, the tradeoff being that the resulting object--known as a magma (which consists of a set and a binary operation, that need not satisfy any properties other than closure)--is simply too general to be interesting. On the other extreme, it is possible to abstract out practically no properties, which allows for many results to be found, but the resulting object (the usual number systems) is too specific to solve more general problems.

Most of abstract algebra is dedicated to objects that have a reasonable balance between generality and structure, most notably groups and rings (discussed in more detail below) in which most of the basic properties of arithmetic are maintained, but their specifics are left free. Still, some higher levels of abstraction are occasionally useful; quasigroups , for instance, are related to Latin squares , and monoids are often used in computer science and are simple examples of categories .

Main article: Group theory The possible moves on a Rubik's cube form a (very large) group . Group theory is useful as an abstract notion of symmetry , which makes it applicable to a wide range of areas: the relationship between the roots of a polynomial (as in Galois theory ) and the solution methods to the Rubik's cube are both prominent examples.

Informally, a group is a set equipped with a binary operation \(\circ\), so that operating on any two elements of the group also produces an element of the group. For example, the integers form a group under addition, and the nonzero real numbers form a group under multiplication. The \(\circ\) operation needs to satisfy a number of properties analogous to the ones it satisfies for these "normal" number systems: it should be associative (which essentially means that the order of operations doesn't matter), and there should be an identity element (0 in the first example above, and 1 in the second). More formally, a group is a set equipped with an operation \(\cdot\) such that the following axioms hold; note that \(\cdot\) does not necessarily refer to multiplication; rather, it should be viewed as a function on two variables (indeed, \(\cdot\) can even refer to addition):

Group Axioms 1) Associativity. For any \(x, y, z \in G \), we have \( (x \cdot y) \cdot z = x \cdot (y \cdot z) \). 2) Identity. There exists an \( e \in G \), such that \( e \cdot x = x \cdot e = x \) for any \(x \in G \). We say that \(e\) is an identity element of \(G\). 3) Inverse. For any \(x \in G\), there exists a \(y \in G\) such that \(x \cdot y = e = y \cdot x \). We say that \(y\) is an inverse of \(x\).

It is also worth noting the closure axiom for emphasis, as it is important to verify closure when working with subgroups (groups contained entirely within another):

4) Closure. For any \(x, y \in G \), \(x*y \) is also in \(G\).

Additional examples of groups include

• \(\mathbb{Z}_n\), the set of integers \(\{0, 1, \ldots, n-1\}\) with the operation addition modulo \(n\)
• \(S_n\), the set of permutations of \(\{1, 2, \ldots, n\}\) with the operation of composition .

\(S_3\) is worth special note as an example of a group that is not commutative , meaning that \(a \cdot b = b \cdot a\) does not generally hold. Formally speaking, \(S_3\) is nonabelian (an abelian group is one in which the operation is commutative). When the operation is not clear from context, groups are written in the form \((\text{set}, \text{op})\); e.g. the nonzero reals equipped with multiplication can be written as \((\mathbb{R}^*, \cdot)\).

Much of group theory (and abstract algebra in general) is centered around the concept of a group homomorphism , which essentially means a mapping from one group to another that preserves the structure of the group. In other words, the mapping of the product of two elements should be the same as the product of the two mappings; intuitively speaking, the product of two elements should not change under the mapping. Formally, a homomorphism is a function \(\phi: G \rightarrow H\) such that

\[\phi(g_1) \cdot_H \phi(g_2) = \phi(g_1 \cdot_G g_2),\]

where \(\cdot_H\) is the operation on \(H\) and \(\cdot_G\) is the operation on \(G\). For example, \(\phi(g) = g \pmod n\) is an example of a group homomorphism from \(\mathbb{Z}\) to \(\mathbb{Z}_n\). The concept of potentially differing operations is necessary; for example, \(\phi(g)=e^g\) is an example of a group homomorphism from \((\mathbb{R},+)\) to \((\mathbb{R}^{*},\cdot)\).

Main article: Ring theory

Rings are one of the lowest level of abstraction, essentially obtained by overwriting the addition and multiplication functions simultaneously (compared to groups, which uses only one operation). Thus a ring is--in some sense--a combination of multiple groups, as a ring can be viewed as a group over either one of its operations. This means that the analysis of groups is also applicable to rings, but rings have additional properties to work with (the tradeoff being that rings are less general and require more conditions).

The definition of a ring is similar to that of a group, with the extra condition that the distributive law holds as well:

A ring is a set \( R \) together with two operations \( + \) and \( \cdot \) satisfying the following properties (ring axioms): (1) \( R \) is an abelian group under addition. That is, \( R\) is closed under addition, there is an additive identity (called \( 0 \)), every element \(a\in R\) has an additive inverse \(-a\in R \), and addition is associative and commutative. (2) \( R \) is closed under multiplication, and multiplication is associative: \(\forall a,b\in R\quad a.b\in R\\ \forall a,b,c\in R\quad a\cdot ( b\cdot c ) =( a\cdot b ) \cdot c.\) (3) Multiplication distributes over addition: \(\forall a,b,c\in R\\ a\cdot \left( b+c \right) =a\cdot b+a\cdot c\quad \text{ and }\quad \left( b+c \right) \cdot a=b\cdot a+c\cdot a.\) A ring is usually denoted by \(\left( R,+,. \right) \) and often it is written only as \(R\) when the operations are understood.

For example, the integers \(\mathbb{Z}\) form a ring, as do the integers modulo \(n\) \((\)denoted by \(\mathbb{Z}_n).\) Less obviously, the square matrices of a given size also form a ring; this ring is noncommutative. Commutative ring theory, or commutative algebra, is much better understood than noncommutative rings are.

As in groups, a ring homomorphism can be defined as a mapping preserving the structure of both operations.

Rings are used extensively in algebraic number theory , where " integers " are reimagined as slightly different objects (for example, Gaussian integers ), and the effect on concepts such as prime factorization is analyzed. Of particular interest is the fundamental theorem of arithmetic , which involves the concept of unique factorization; in other rings, this may not hold, such as

\[6 = 2 \cdot 3 = \big(1+\sqrt{-5}\big)\big(1-\sqrt{-5}\big).\]

Theory developed in this field solves problems ranging from sum of squares theorems to Fermat's last theorem .

Abstract algebra also has heavy application in physics and computer science through the analysis of vector spaces . For example, the Fourier transform and differential geometry both have vector spaces as their underlying structures; in fact, the Poincare conjecture is (roughly speaking) a statement about whether the fundamental group of a manifold determines if the manifold is a sphere.

Related to vector spaces are modules , which are essentially identical to vector spaces but defined over a ring rather than over a field (and are thus more general). Modules are heavily related to representation theory , which views the elements of a group as linear transformations of a vector space; this is desirable to make an abstract object (a group) somewhat more concrete, in the sense that the group is better understood by translating it into a well-understood object in linear algebra (as matrices can be viewed as linear transformations, and vice versa).

The relationships between various algebraic structures are formalized using category theory .

• Group Theory Introduction

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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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MATH 2101 Abstract Algebra I

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This page is a draft and is under active development. 

• Pamini Thangarajah
• Mount Royal University

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Course Description

This course is an introduction to abstract algebra with applications in cryptography. Topics studied include groups and rings, polynomial arithmetic and the division algorithm, and congruencies.

Prerequisite: Math 1203 Linear Algebra for Scientists and Engineers with a grade of C- or higher.

The content was created by Pamini Thangarajah and her winter 2019 MATH 2101 class under the supervision of Pamini Thangarajah and used in MATH 2101 at Mount Royal University. Updated in winter 2021 & then Fall 2023.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .

References and Suggested readings:

1. Dummit, D. S., & Foote, R. M. (2004).  Abstract algebra  (Vol. 3). Hoboken: Wiley.

2. Gallian, J. (2021).  Contemporary abstract algebra . Chapman and Hall/CRC.

3. Judson, T. W. (2020).  Abstract algebra: theory and applications . ( http://abstract.ups.edu/aata/aata.html )

4.  Nicholson, W. K. (2012).  Introduction to abstract algebra . John Wiley & Sons.

5. Paulin, A. (2021). Introduction to Abstract Algebra (Math 113).  University of California, Berkeley, available online . ( https://math.berkeley.edu/~apaulin/A...actAlgebra.pdf )

• Table of Contents
• 0.0: Introduction to proofs
• 0.2: Relations
• 0.3: Functions
• 0.5: Proof Templates
• 0E: Exercises
• 1.1: Division Algorithm
• 1.2: Greatest common divisor and least common multiple
• 1.3: Primes
• 1E: Exercises
• 2.1: Introduction to Group
• 2.2: Properties of Group Elements
• 2.3: Subgroups
• 2.4: Introduction to Cyclic groups
• 2E: Exercises
• 3.1: Symmetric Groups
• 3.2: Alternating Groups
• 3.3: Dihedral Groups (Group of Symmetries)
• 3E: Exercises
• 4.1: Cosets and Lagranges Theorem
• 4.2: Normal Groups and Factor Groups
• 4.3 : Homomorphisms
• 4.4: On Special Groups
• 4.5: Isormormophism theorems
• 4.6: Classification Groups
• 4E: Exercises
• 6.1: Introduction to Rings
• 6.2: Ring Homomorphisms
• 6E: Exercises

• Detailed Licensing

Math 113: Abstract algebra

Uc berkeley, fall 2010, course goals, notes on proofs, homework policy.

• Homework assignments

Exams and grading

• Short syllabus
• Long syllabus

A group is, roughly, a set with one "binary operation" on it satisfying certain axioms which we will learn about. Examples of groups include the integers with the operation of addition, the nonzero real numbers with the operation of multiplication, and the invertible n by n matrices with the operation of matrix multiplication. But groups arise in many other diverse ways. For example, the symmetries of an object in space naturally comprise a group. After studying many examples of groups, we will develop some general theory which concerns the basic principles underlying all groups.

A ring is, roughly, a set with two binary operations on it satisfying certain properties which we will learn about. An example is the integers with the operations of addition and multiplication. Another example is the ring of polynomials. A field is a ring with certain additional nice properties, such as the rational and real numbers. At the end of the course we will have built up enough machinery to prove that one cannot trisect a sixty degree angle using a ruler and compass.

In addition to the specific topics we will study, which lie at the foundations of much of higher mathematics, an important goal of the course is to expand facility with mathematical reasoning and proofs in general, as a transition to more advanced mathematics courses, and for logical thinking outside of mathematics as well. I am hoping that you already have some familiarity with proofs from Math 55 or a similar course. If not, the following might be helpful:

Most of the lectures will correspond to particular sections of the book (indicated in the syllabus below), and studying these sections should be very helpful for understanding the material. However, please note that in class I will often present material in a different order or from a different perspective than that of the book. We will also occasionally discuss topics which are not in the book at all. Thus it is important to attend class and, since you shouldn't expect to understand everything right away , to take good notes.

There are many other algebra texts out there, and you might try browsing through these for some additional perspectives. (Bear in mind that Fraleigh is an "entry-level" text, so many other algebra books will be too hard at this point; but after this course you should be prepared to start exploring these. There is a vast world of algebra out there!)

In addition, the math articles on wikipedia have gotten a lot better than they used to be, and much useful information related to this course can be found there. However you shouldn't blindly trust anything you read on the internet, and keep in mind that wikipedia articles tend to give brief summaries rather than the detailed explanations that are needed for proper understanding.

Other references:

Algebra: Abstract and Concrete , Edition 2.5, Frederick M. Goodman

Applied Abstract Algebra ,  Rudolf Lidl and Gunter Pilz: this book gives applications of abstract algebra, but is a second course (you can read online through the library). Basic Algebra, Groups, Rings and Fields , P. M. Cohn

Abstract Algebra , Paul Garrett (chapter 01 has some background on number theory)

Study tips (for any upper division math course)

• It is essential to thoroughly learn the definitions of the concepts we will be studying. You don't have to memorize the exact wording given in class or in the book, but you do need to remember all the little clauses and conditions. If you don't know exactly what a UFD is, then you have no hope of proving that something is or is not a UFD. In addition, learning a definition means not just being able to recite the definition from memory, but also having an intuitive idea of what the definition means, knowing some examples and non-examples, and having some practical skill in working with the definition in mathematical arguments.
• In the same way it is necessary to learn the statements of the theorems that we will be proving.
• It is not necessary to memorize the proofs of theorems. However the more proofs you understand, the better your command of the material will be. When you study a proof, a useful aid to memory and understanding is to try to summarize the key ideas of the proof in a sentence or two . If you can't do this, then you probably don't yet really understand the proof. 
• The material in this course is cumulative and gets somewhat harder as it goes along, so it is essential that you do not fall behind .
• If you want to really understand the material, the key is to ask your own questions . Can I find a good example of this? Is that hypothesis in that theorem really necessary? What happens if I drop it? Can I find a different proof using this other strategy? Does that other theorem have a generalization to the noncommutative case? Does this property imply that property, and if not, can I find a counterexample? Why is that condition in that definition there? What if I change it this way? This reminds me of something I saw in linear algebra; is there a direct connection?
• If you get stuck on any of the above, you are welcome to come to my office hours . I am happy to discuss this stuff with you. Usually, the more thought you have put in beforehand, the more productive the discussion is likely to be.

When preparing your homework, please keep the following in mind:

1) You are encouraged to discuss the homework problems with your classmates. Mathematics can be a fun social activity! Perhaps the best way to learn is to think hard about a problem on your own until you get really stuck or solve it, then ask someone else how they thought about it. However, when it comes time to write down your solutions to hand in, you must do this by yourself , in your own words, without copying or looking at someone else's paper. If you obtain help from another student with a problem, write this down on your homework. This will not affect your grade, unless it is clear that you have copied verbatim - the final answer should be written in your own words. It is important to get in the habit of citing your sources, which may include your colleagues (otherwise, you are plagiarizing)!

2) All answers should be written in complete, grammatically correct English sentences which explain the logic of what you are doing, with mathematical symbols and equations interspersed as appropriate. For example, instead of writing "x^2 = 4, x = 2, x = -2", write "since x^2 = 4, it follows that x = 2 or x = -2." Otherwise your proof will be unreadable and will not receive credit . Results of calculations and answers to true/false questions etc. should always be justified. Proofs should be complete and detailed. The proofs in the book provide good models; but when in doubt, explain more details. Avoid phrases such as "it is easy to see that"; often what follows such a phrase is actually a tricky point that needs justifiction, or even false. You can of course cite theorems that we have already proved in class or from the book.

Reading Assignments (from Fraleigh)

• 8/31: Read Notes on proofs, Suggestions for writing mathematics, and Sections 0, 1 of Fraleigh
• 9/2: Read Sections 2 and 3
• 9/7: Read Sections 3 and 4
• 9/9: Read Sections 5 and 6
• 9/14: Read Sections 6 and 7
• 9/16: Read Sections 8 and 9
• 9/21: Read Sections 8 and 9
• 9/23: Read Sections 9 and 10
• 9/28: Read Sections 10 and 11
• 9/30: Read Sections 11 and 12
• 10/5: Read Section 12
• 10/7: Midterm 1
• 10/12: Read Sections 12 and 13
• 10/14: Read Section 13
• 10/19: Read Section 14
• 10/21: Read Section 15
• 10/26: Read Section 18
• 10/28: Read Section 19
• 11/2: Review
• 11/4: Midterm 2
• 11/9: Read Section 20
• 11/11: Veteran's day, no class
• 11/16: Read Section 21
• 11/18: Read Section 22
• 11/23: Read Section 23
• 11/25: Thanksgiving, no class
• 11/30: Read Section 26
• 12/2: Read Section 27

Homework assignments:

• Homework 1 due 9/9 (at the beginning of class): Section 0, #29-34*; Section 2, #5,6,9,13,23,37, *explain for each problem from Section 0 which property of equivalence relations hold, and which do not, with justification.  Homework 1 solutions (cut and paste into a web browser): http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_1_sol.pdf
• Homework 2 due 9/16: Section 3 # 2, 4, 6,  27, 33 ; Section 4 # 9, 10, 37; Section 5 # 13, 23, 49; Section 6 # 22, 48. Homework 2 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_2_sol.pdf
• Homework 3 due 9/23: Section 7 # 11, 18, Section 8 # 3, 8,  21, 45, 46. Homework 3 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_3_sol.pdf
• Homweork 4 due 9/30: Section 9 # 7, 13, 16, 29, Section 10, # 6, 7, 28, 29, 39. Homework 4 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_4_sol.pdf
• Solutions to Midterm 1
• Homework 5 due 10/14: Section 11, # 10, 18, 20, 30, 47, Section 12 #8, 16, 24-30, 38, 40. Homework 5 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_5_sol.pdf
• Homework 6 due 10/21: Section 13, # 2, 6, 8, 20, 36, 37, 44, 50  Section 14 # 3, 6, 27, 31, 32,  33. Homework 6 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_6_sol.pdf
• Homework 7 due 10/28: Section 15, # 3, 7, 13, 36, 37, Section 18 # 9, 15, 20, 25, 37, 41. Homework 7 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_7_sol.pdf
• No homework due 11/11 because of Veteran's day
• Solutions to Midterm 2 Section 2: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_mt_2_Section_2_solutions.pdf
• Solutions to Midterm 2 Section 4: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_mt_2_Section_4_solutions.pdf
• Homework 8 due 11/18: Section 19,  #3, 9, 25, 29; Section 20 #6, 7, 14,  27, 28; Section 21 #1, 2, 5, 7. Homework 8 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_8_sol.pdf
• Homework 9 due 12/2: Section 22, # 5, 14, 25, 30; Section 23 #3, 12, 17, 28, 34; Section 26 #3, 17, 18, 20; Homework 9 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_9_sol.pdf

There will be no makeup exams. However you can miss one midterm without penalty, as explained in the grading policy below.

There is no regrading unless there is an egregious error such as adding up the points incorrectly. Every effort is made to grade all exams according to the same standards, so regrading one student's exam would be unfair to everyone else.

The course grade will be determined as follows: homework 20%, midterms 20% each, final 60%, lowest exam score -20%. All grades will be curved to a uniform scale before being averaged.

Syllabus (short version)

• Preliminaries . We will begin with a review of some essential preliminaries, including sets, functions, relations, induction, and some very basic number theory. You have probably already seen this material in Math 55 or elsewhere, so the review will be brief. Some of this material is in section 0 of the book, some is scattered throughout random later sections, some is in the above notes on proofs, and some is in none of the above.
• Groups . We will learn a lot about groups, starting with the detailed study of a slew of examples, and then proceeding to some important general principles. We will cover most of Parts I, II, and III of the book. We will consider a few examples which are not in the book, such as symmetry groups of polyhedra and wallpaper groups. We will mostly skip the advanced group theory in Part VII, aside from stating a couple of the results. (You can learn some of this material in Math 114.) We will completely skip Part VIII on group theory in topology; this material is best learned in a topology course such as Math 142.
• Ring theory and polynomials . Next we will learn about rings. We will pay particular attention to rings of polynomials, which are very important e.g. in algebraic geometry. We will cover most of Parts IV, V, and IX.
• Elements of field theory . Finally, after reviewing some notions from linear algebra in a more general setting, we will learn the basics of fields, from Part VI of the book. We will develop enough machinery to prove that one cannot trisect a sixty degree angle with a ruler and compass. We will not have time for the more advanced field theory in Part X, including the insolvability of the quintic; this is covered in Math 114.

Syllabus (long version): 

 8/26, 8/31: about the course. brief review of preliminaries: sets, functions, injections, surjections, bijections. equivalence relations and modular arithmetic. proof by induction, strong induction, and the well ordering principle. the division theorem. greatest common divisors, the euclidean algorithm, and solving ax = b (mod n). the fundamental theorem of arithmetic. [0,notes on proofs]. for fun: play the game set.

• 9/2: Binary operations. Isomorphism of binary structures. Using structural properties to show that binary structures are not isomorphic. [Sections 2,3]
• 9/7: Groups: definition, many examples [Section 4]
• 9/9: Subgroups, cyclic groups [Sections 4,5]
• 9/14: cyclic groups [Section 6]
• 9/16: Cayley graphs [Sections 7]
• 9/21: permutation groups [Sections 8, 9]
• 9/23: orbits, alternating groups, cosets [9,10]
• 9/28: Lagrange's theorem [10]
• 9/30: Product groups [11]
• 10/5: review
• 10/12: plane isometries,  frieze and wallpaper groups [12]  Wallpaper patterns
• 10/14: homomorphisms [13]
• 10/19: factor groups [14]
• 10/21: factor group computations and simple groups [15]
• 10/26: rings and fields [18]
• 10/28: integral domains [19]
• 11/9: Fermat's and Euler's theorems [20]
• 11/16: Field of quotients [21]
• 11/18: Polynomial rings [22]
• 11/23: Factorization of polynomials [23]
• 11/30: Homomorphisms and factor rings [26]
• 12/2: Prime and maximal ideals [27]

MATH3311 - Abstract Algebra I

MATH 3311 Abstract Algebra I (3 semester credit hours) Groups, rings, fields, vector spaces modules, linear transformations, and Galois theory. Prerequisites: A grade of at least a C- in either MATH 2415 or in MATH 2419 or equivalent and a grade of at least C- in MATH 2418 or equivalent. (3-0) S

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The Algebra Problem: How Middle School Math Became a National Flashpoint

Top students can benefit greatly by being offered the subject early. But many districts offer few Black and Latino eighth graders a chance to study it.

By Troy Closson

From suburbs in the Northeast to major cities on the West Coast, a surprising subject is prompting ballot measures, lawsuits and bitter fights among parents: algebra.

Students have been required for decades to learn to solve for the variable x, and to find the slope of a line. Most complete the course in their first year of high school. But top-achievers are sometimes allowed to enroll earlier, typically in eighth grade.

The dual pathways inspire some of the most fiery debates over equity and academic opportunity in American education.

Do bias and inequality keep Black and Latino children off the fast track? Should middle schools eliminate algebra to level the playing field? What if standout pupils lose the chance to challenge themselves?

The questions are so fraught because algebra functions as a crucial crossroads in the education system. Students who fail it are far less likely to graduate. Those who take it early can take calculus by 12th grade, giving them a potential edge when applying to elite universities and lifting them toward society’s most high-status and lucrative professions.

But racial and economic gaps in math achievement are wide in the United States, and grew wider during the pandemic. In some states, nearly four in five poor children do not meet math standards.

To close those gaps, New York City’s previous mayor, Bill de Blasio, adopted a goal embraced by many districts elsewhere. Every middle school would offer algebra, and principals could opt to enroll all of their eighth graders in the class. San Francisco took an opposite approach: If some children could not reach algebra by middle school, no one would be allowed to take it.

The central mission in both cities was to help disadvantaged students. But solving the algebra dilemma can be more complex than solving the quadratic formula.

New York’s dream of “algebra for all” was never fully realized, and Mayor Eric Adams’s administration changed the goal to improving outcomes for ninth graders taking algebra. In San Francisco, dismantling middle-school algebra did little to end racial inequities among students in advanced math classes. After a huge public outcry, the district decided to reverse course.

“You wouldn’t think that there could be a more boring topic in the world,” said Thurston Domina, a professor at the University of North Carolina. “And yet, it’s this place of incredibly high passions.”

“Things run hot,” he said.

In some cities, disputes over algebra have been so intense that parents have sued school districts, protested outside mayors’ offices and campaigned for the ouster of school board members.

Teaching math in middle school is a challenge for educators in part because that is when the material becomes more complex, with students moving from multiplication tables to equations and abstract concepts. Students who have not mastered the basic skills can quickly become lost, and it can be difficult for them to catch up.

Many school districts have traditionally responded to divergent achievement levels by simply separating children into distinct pathways, placing some in general math classes while offering others algebra as an accelerated option. Such sorting, known as tracking, appeals to parents who want their children to reach advanced math as quickly as possible.

But tracking has cast an uncomfortable spotlight on inequality. Around a quarter of all students in the United States take algebra in middle school. But only about 12 percent of Black and Latino eighth graders do, compared with roughly 24 percent of white pupils, a federal report found .

“That’s why middle school math is this flashpoint,” said Joshua Goodman, an associate professor of education and economics at Boston University. “It’s the first moment where you potentially make it very obvious and explicit that there are knowledge gaps opening up.”

In the decades-long war over math, San Francisco has emerged as a prominent battleground.

California once required that all eighth graders take algebra. But lower-performing middle school students often struggle when forced to enroll in the class, research shows. San Francisco later stopped offering the class in eighth grade. But the ban did little to close achievement gaps in more advanced math classes, recent research has found.

As the pendulum swung, the only constant was anger. Leading Bay Area academics disparaged one another’s research . A group of parents even sued the district last spring. “Denying students the opportunity to skip ahead in math when their intellectual ability clearly allows for it greatly harms their potential for future achievement,” their lawsuit said.

The city is now back to where it began: Middle school algebra — for some, not necessarily for all — will return in August. The experience underscored how every approach carries risks.

“Schools really don’t know what to do,” said Jon R. Star, an educational psychologist at Harvard who has studied algebra education. “And it’s just leading to a lot of tension.”

In Cambridge, Mass., the school district phased out middle school algebra before the pandemic. But some argued that the move had backfired: Families who could afford to simply paid for their children to take accelerated math outside of school.

“It’s the worst of all possible worlds for equity,” Jacob Barandes, a Cambridge parent, said at a school board meeting.

Elsewhere, many students lack options to take the class early: One of Philadelphia’s most prestigious high schools requires students to pass algebra before enrolling, preventing many low-income children from applying because they attend middle schools that do not offer the class.

In New York, Mr. de Blasio sought to tackle the disparities when he announced a plan in 2015 to offer algebra — but not require it — in all of the city’s middle schools. More than 15,000 eighth graders did not have the class at their schools at the time.

Since then, the number of middle schools that offer algebra has risen to about 80 percent from 60 percent. But white and Asian American students still pass state algebra tests at higher rates than their peers.

The city’s current schools chancellor, David Banks, also shifted the system’s algebra focus to high schools, requiring the same ninth-grade curriculum at many schools in a move that has won both support and backlash from educators.

And some New York City families are still worried about middle school. A group of parent leaders in Manhattan recently asked the district to create more accelerated math options before high school, saying that many young students must seek out higher-level instruction outside the public school system.

In a vast district like New York — where some schools are filled with children from well-off families and others mainly educate homeless children — the challenge in math education can be that “incredible diversity,” said Pedro A. Noguera, the dean of the University of Southern California’s Rossier School of Education.

“You have some kids who are ready for algebra in fourth grade, and they should not be denied it,” Mr. Noguera said. “Others are still struggling with arithmetic in high school, and they need support.”

Many schools are unequipped to teach children with disparate math skills in a single classroom. Some educators lack the training they need to help students who have fallen behind, while also challenging those working at grade level or beyond.

Some schools have tried to find ways to tackle the issue on their own. KIPP charter schools in New York have added an additional half-hour of math time to many students’ schedules, to give children more time for practice and support so they can be ready for algebra by eighth grade.

At Middle School 50 in Brooklyn, where all eighth graders take algebra, teachers rewrote lesson plans for sixth- and seventh-grade students to lay the groundwork for the class.

The school’s principal, Ben Honoroff, said he expected that some students would have to retake the class in high school. But after starting a small algebra pilot program a few years ago, he came to believe that exposing children early could benefit everyone — as long as students came into it well prepared.

Looking around at the students who were not enrolling in the class, Mr. Honoroff said, “we asked, ‘Are there other kids that would excel in this?’”

“The answer was 100 percent, yes,” he added. “That was not something that I could live with.”

Troy Closson reports on K-12 schools in New York City for The Times. More about Troy Closson

Three Juniors Named Faculty Scholars

Awards are the highest honor bestowed by faculty to undergraduates

Three undergraduate students whose research shows their deep expertise on number theory, 19 th century American history and analysis of poetry were honored with Faculty Scholar Awards, the highest honor bestowed by university faculty on undergraduates.

Presented through the Academic Council, the 2024 winners are Sarah Konrad, Arielle Stern and Marie-Hélène Tomé. The award was established to highlight students who are likely to pursue a scholarly career and already have established a record of research and independent study that has impressed their faculty mentors.

The three were selected by a faculty committee chaired by Sheryl Broverman, professor of the practice of biology. The committee received 27 strong nominations from departments across the undergraduate program.

“As always this is a very challenging process, as so many of our students are doing exceptional, original research,” Broverman said. “The three Faculty Scholar recipients all impressed the committee with their in-depth knowledge of a field and their ability to communicate it to non-experts in the field.  From explaining the world via math proofs or the human condition via poetry to the history of slaveholding by women, the committee was deeply impressed by Marie- Hélène, Arielle and Sarah.”

Sarah Konrad, Department of History

With an interest in both history and law, Sarah Konrad has already produced a collection of original research that explores how the law affects social, cultural and political aspects of public life. Like a good historian, she has written several complicated portraits of women in 19th century America, living with limited legal rights but still finding ways to exercise power, and affecting issues of race.

One large project was close to home. Working with history professors Thavolia Glymph and Robert Korstad as part of the Duke Institutional History Project, Konrad dove into the early days of Trinity College to explore the relationships between the wives of the college’s Board of Trustees and how they benefited from the institution of slavery.

Konrad found close family ties between the board wives and their husbands, “creating a familial connection that pervaded the bonds of the academic administration,” she said. These ties were strengthened by slavery, as the wives often brought enslaved people with them into the marriage, which grew the economic status of their husbands. The research will be included as a chapter in a forthcoming book from the institutional history project published by Duke University Press.

In a second research project, under the supervision of professors Juliana Barr and Sarah Deutsch, Konrad explored stories of Cherokee women who owned enslaved people. She will complete this next year as her honors thesis.

“Sarah Konrad is an extraordinary scholar – an indefatigable researcher, creative both in how she ferrets out sources and how she makes sense of them, as well in the even more important area of how she comes up with and formulates a question,” said Sarah Deutsch, professor emerita of history. “Her excitement is contagious.”

Konrad says she hopes to continue this research following graduation in 2025 and will seek a joint J.D./Ph.D. degree. “With lifelong research efforts, I hope to contribute to historical and legal scholarship that bridges strict divisions of past and present to show how law has been formed by historical processes, and yet it can still be used as a tool of justice,” she said.

Arielle Stern – Department of English

To Arielle Stern, poetry is the place where the known and the unknown are placed together, where words “function to elucidate hidden and incomprehensible meanings, but do not erase the murkiness of the shadows that linger.”

That richness of meaning and language has long attracted Stern and has led to several research projects praised by Duke faculty members. In a graduate-level course on 20 th century French theory, Stern considered historical memory in post-WWII poetry, particularly related to the Holocaust. The paper, which she was invited to present at a research symposium, explored ethical and literary questions of how to write about atrocity.

“Intrigued by the pervasiveness of absence in the aftermath of WWII, I was compelled to probe deeper into the question of how to portray extreme erasure, to both preserve memory and to acknowledge the gaps that constitute the difficulty of such a task,” she said.

An English and Romance Studies double major, Stern also has focused on Wallace Stevens and studying his rich poetry through the lens of Stevens’ interest in the French linguistic and philosophical traditions. A poet herself, this study has also benefited her own writings.

Stern will write a senior thesis exploring a number of modernist poets’ last books that dwell at the horizon of death – the horizon of unknowing – to understand a state defined by its distinct uncertainty. To draw on her interest in French thinkers, she will also look to French philosophers Jacques Derrida and Maurice Blanchot among others to guide the inquiry.

 “The first of Arielle’s numerous critical gifts is the quality of her alertness to the poem,” said Joseph Donahue, professor of the practice of English, who directed some of her study of Stevens. “She approaches the page with allegiance to her already deeply schooled sophistication, but, always first, she sees and hears for herself what is going on in the poem and finds her own way to imaginatively enter into the world the text proposes.

“She expertly moves into unfamiliar terrain and makes it her own, even when the terrain is most forbidding, and so her interest in the poetics of death, in the great tradition in world literature of poetry written at the threshold of the abyss, at the absolute limit of what can be known and felt. Where else would such a curious and capable imagination as that possessed by Arielle Stern be spending its time?”

After graduation, Stern hopes to study for a Ph.D. in English Literature focusing on 20th and 21st century poetry and poetics. “The study of poetry itself is that of making sense of the

unknown and the purposefully obscured, an exercise that rejects the denial of erasure and brings absence to light, which I intend to do in my future pursuits, both when writing poetry and in a scholarly career,” Stern said.

Marie-Hélène Tomé – Department of Mathematics

In number theory, L-functions package important arithmetic information about the mathematical objects they are associated to. L-functions, of which the Riemann zeta function is a particular example, are the subject of many of the most challenging unresolved conjectures in mathematics.

This year, Marie-Hélène Tomé effectively answered an open L-function conjecture made in 1920 by the German Erich Hecke.

Tomé is a recipient of a 2024 Goldwater Scholarship, a nationally competitive award for students in mathematics, natural sciences and engineering. Part of her recognition came from her work on Hecke L-functions and their special values. Under the guidance of Professor Ken Ono at the University of Virginia, she studied the work of Japanese mathematician Takuro Shintani, who provides formulas for the class number, an important arithmetic invariant associated with a number field.

Shintani’s formulas answer Hecke’s conjecture for biquadratic extensions (n = 2). Building on Shintani’s work, Tomé derived finite formulas for relative quadratic extensions of fields F of arbitrary degree n over the rational numbers, together with methods to explicitly compute the inputs to these formulas. Her work gives an effective affirmative answer to Hecke’s conjecture for arbitrary degree n for a certain class of extensions. Her solutions presented a novel method to make the difficult calculations involved in the conjecture and opens possibilities for the solution of other similar mathematical questions.

This work was presented at the 2024 Joint Mathematics Meetings and resulted in a single author paper that will soon appear in the Journal of Number Theory. While Tomé has long been interested in mathematics, she became interested in number theory while taking abstract algebra with Duke Professor Robert Calderbank. A guest lecture by Professor Lillian Pierce piqued her interest in the rich intersection of algebraic and analytic number theory. While participating in the 2023 REU (Research Experience for Undergraduates) in number theory at the University of Virginia, her interest grew into a deep passion.

In addition, Tomé has completed an independent study with Lillian Pierce, professor of mathematics, on the Weil-Deligne bound, which has diverse applications in analytic number theory. Under the guidance of Professor Pierce, Tomé wrote an expository paper on Schmidt’s proof of the Weil-Deligne bound. She will complete an honors thesis on topics in algebraic number theory with Professor Samit Dasgupta.

“As mathematics students become independent mathematicians, they learn to be very skeptical, in the best possible sense,” Pierce said. “Mathematicians look to understand the precise reasons that a proof method works, both to make sure that all the details are correct, and also to understand the limitations of the method. Understanding these limitations is critical to being able to go onward with original research.

“Marie-Hélène worked to learn this material with the skeptical style of an independent mathematician. She left no stone unturned while she studied multiple research papers (in multiple languages) to develop a complete understanding of this important result.”

After graduation, Tomé intends to pursue a Ph.D. in pure mathematics and conduct research at the intersection of algebraic and analytic number theory. She hopes to become a professor of mathematics at a research university where she can combine her love of teaching with her passion for research.

“My research experiences in number theory crystallized my career goal of becoming a number theorist,” she said. “My natural curiosity has both informed my previous research in mathematics and the mathematician I hope to become. As a mathematician, I see myself continually learning new mathematics to weave diverse areas into my research and apply techniques from other fields of mathematics to solve questions in number theory. Something beautiful and mysterious lying within the mathematics of number theory calls to me, and I cannot refuse that call.”

Learning 3D Matrix Algebra Using Virtual and Physical Manipulatives: Qualitative Analysis of the Efficacy of the AR-Classroom

• Conference paper
• First Online: 01 June 2024
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• Samantha D. Aguilar   ORCID: orcid.org/0000-0002-1010-6082 26 ,
• Heather Burte   ORCID: orcid.org/0000-0002-9623-4375 27 ,
• James Stautler   ORCID: orcid.org/0009-0007-6192-3174 26 ,
• Sadrita Mondal   ORCID: orcid.org/0009-0008-0594-6686 26 ,
• Chengyuan Qian   ORCID: orcid.org/0009-0008-0974-0992 26 ,
• Uttamasha Monjoree   ORCID: orcid.org/0000-0001-5246-3582 26 ,
• Philip Yasskin   ORCID: orcid.org/0000-0003-3103-8127 26 ,
• Jeffrey Liew   ORCID: orcid.org/0000-0002-0784-8448 26 ,
• Dezhen Song   ORCID: orcid.org/0000-0002-2944-5754 26 , 28 &
• Wei Yan   ORCID: orcid.org/0000-0003-1092-2474 26  

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14724))

Included in the following conference series:

• International Conference on Human-Computer Interaction

The AR-Classroom application aims to teach three-dimensional (3D) geometric rotations and their underlying mathematics using virtual and physical manipulatives. In an efficacy experiment, undergraduates completed six 3D matrix algebra rotation questions and were assigned to interact with virtual (N = 20) or physical (N = 20) manipulatives in the AR-Classroom. While completing these rotation questions, researchers documented the participants’ reported thoughts, feelings, and perceptions (i.e., qualitative data). A thematic analysis of participants’ reports revealed four prevalent themes regarding participants’ learning experience: (1) Difficulty using traditional methods, (2) Reliance on resources, (3) Pattern recognition, and (4) Developing an understanding of 3D matrix algebra. Participants struggled to complete rotation matrices when only using information from the question and the model; when unsure how to solve the matrix, participants utilized any available resources. Moreover, participants could identify similarities among matrices, demonstrated after using AR-Classroom repeatedly. The findings indicate that the AR-Classroom may aid students in improving their mathematical skills. Suggestions for future research on the AR-Classroom and efficacy experiments are discussed.

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Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. 2119549. We appreciate the support from our undergraduate learning and assessment research team, Adalia Sedigh, Grace Girgenti, Hana Syed, and Megan Sculley.

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Samantha D. Aguilar, James Stautler, Sadrita Mondal, Chengyuan Qian, Uttamasha Monjoree, Philip Yasskin, Jeffrey Liew, Dezhen Song & Wei Yan

Carnegie Mellon University, Pittsburg, PA, USA

Heather Burte

Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates

Dezhen Song

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Aguilar, S.D. et al. (2024). Learning 3D Matrix Algebra Using Virtual and Physical Manipulatives: Qualitative Analysis of the Efficacy of the AR-Classroom. In: Zaphiris, P., Ioannou, A. (eds) Learning and Collaboration Technologies. HCII 2024. Lecture Notes in Computer Science, vol 14724. Springer, Cham. https://doi.org/10.1007/978-3-031-61691-4_1

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