abstract algebra phd

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Algebra PhD

Awards: PhD

Study modes: Full-time, Part-time

Funding opportunities

Programme website: Algebra

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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:

abstract algebra Recently Published Documents

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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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Abstract Algebra

• Choi, Evelyn, Shannon Fan, Jadon Geathers, and Aoden Teo Masa Toshi. “On the Partition Function of Six Vertex Lattice Models Through Interpolation and Demazure Operators” .
• Mahoney, Dylan, and Bryan Park. “On the Weierstrass p_Function As an Exponential Map” .
• Hao, Hanson. “Investigations on Automorphism Groups of Quantum Stabilizer Codes” .
• Hao, Hanson, Eli Navaro, and Henri Stern. “Irreducibility and Galois Groups of Random Polynomials” .
• Ido, Yuzu, Ian Ruohoniemi, and Matthew Stevens. “Integer Forms and Elliptic Curves” .
• Greicius, Quinn. “Symmetric Homomorphisms of Abelian Varieties” .
• Singh, Jaydeep. “Topics in Lattice Gauge Theory: Behavior of Wilson Loops in the Thermodynamic Limit” .
• Bitz, Jared, and Cory Griffith. “A New Bound and New Techniques for the Erdos–Ginzburg–Ziv Constant for Finite Abelian Groups” .
• Tu, Leslie. “Schubert Calculus on the Grassmanian” .

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

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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

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Abstract Algebra

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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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Abstract Algebra: Theory and Applications (Judson)

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• Thomas W. Judson
• Stephen F. Austin State University via Abstract Algebra: Theory and Applications

Abstract Algebra: Theory and Applications is designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory.

Meeting Time

TH 11:00-12:15am

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M: 3-4pm, T: 4-5pm, W: 4-5pm, Kerchof 213

Class Notes

Homework assigments.

• Homework #1, Due February 7th & Solutions
• Homework #2, Due February 28th: 30, 32, 34a,b, 35, 40a, 47, 49, 50, 52 & Solutions
• Homework #3, Due April 1st
• Homework #4, Due April 22nd: 2, 6, 47, 57-61, 63

Toolkit Stuff

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.

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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, marc a. rieffel, thomas scanlon, vera serganova.

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Abstract Algebra

Welcome to the course web page for the Spring 2022 manifestation of MAT 411: Introduction to Abstract Algebra at Northern Arizona University .

Course Info

Instructor info, what is this course all about.

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. This course is an introduction to abstract algebra. We will spend most of our time studying groups. Group theory is the study of symmetry, and is one of the most beautiful areas in all of mathematics. It arises in puzzles, visual arts, music, nature, the physical and life sciences, computer science, cryptography, and of course, throughout mathematics. This course will cover the basic concepts of group theory, and a special effort will be made to emphasize the intuition behind the concepts and motivate the subject matter. In the last few weeks of the semester, we will also introduce rings and fields.

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. Henri Poincaré

An Inquiry-Based Approach

This is not a lecture-oriented class or one in which mimicking prefabricated examples will lead you to success. You will be expected to work actively to construct your own understanding of the topics at hand, with the readily available help of me and your classmates. Many of the concepts you learn and problems you work will be new to you and ask you to stretch your thinking. You will experience frustration and failure before you experience understanding . This is part of the normal learning process. If you are doing things well, you should be confused at different points in the semester. The material is too rich for a human being to completely understand it immediately. Your viability as a professional in the modern workforce depends on your ability to embrace this learning process and make it work for you.

In order to promote a more active participation in your learning, we will incorporate ideas from an educational philosophy called inquiry-based learning (IBL). Loosely speaking, IBL is a student-centered method of teaching mathematics that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Rather than showing facts or a clear, smooth path to a solution, the instructor guides and mentors students via well-crafted problems through an adventure in mathematical discovery. According to Laursen and Rasmussen (PDF), the Four Pillars of IBL are:

• Students engage deeply with coherent \& meaningful mathematical tasks.
• Students collaboratively process mathematical ideas.
• Instructors inquire into student thinking.
• Instructors foster equity in their design and facilitation choices.

If you want to learn more about IBL, read my blog post titled What the Heck is IBL? .

Don’t fear failure. Not failure, but low aim, is the crime. In great attempts it is glorious even to fail. Bruce Lee

Much of the course will be devoted to students presenting their proposed solutions/proofs on the board and a significant portion of your grade will be determined by how much mathematics you produce. I use the word “produce” because I believe that the best way to learn mathematics is by doing mathematics. Someone cannot master a musical instrument or a martial art by simply watching, and in a similar fashion, you cannot master mathematics by simply watching; you must do mathematics!

In any act of creation, there must be room for experimentation, and thus allowance for mistakes, even failure. A key goal of our community is that we support each other—sharpening each other’s thinking but also bolstering each other’s confidence—so that we can make failure a productive experience. Mistakes are inevitable, and they should not be an obstacle to further progress. It’s normal to struggle and be confused as you work through new material. Accepting that means you can keep working even while feeling stuck, until you overcome and reach even greater accomplishments.

You will become clever through your mistakes. German Proverb

Furthermore, it is important to understand that solving genuine problems is difficult and takes time. You shouldn’t expect to complete each problem in 10 minutes or less. Sometimes, you might have to stare at the problem for an hour before even understanding how to get started. In fact, solving difficult problems can be a lot like the clip from the Big Bang Theory located here .

In this course, everyone will be required to

• read and interact with course notes and textbook on your own;
• write up quality solutions/proofs to assigned problems;
• present solutions/proofs on the board to the rest of the class;
• participate in discussions centered around a student’s presented solution/proof;
• call upon your own prodigious mental faculties to respond in flexible, thoughtful, and creative ways to problems that may seem unfamiliar at first glance.

As the semester progresses, it should become clear to you what the expectations are.

Tell me and I forget, teach me and I may remember, involve me and I learn. Benjamin Franklin

Dana C. Ernst

Mathematics & Teaching

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Abstract algebra : structures and applications

Available online, at the library.

Science Library (Li and Ma)

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Description

Creators/contributors, contents/summary.

• SET THEORY Sets and Functions The Cartesian Product-- Operations-- Relations Equivalence Relations Partial Orders
• NUMBER THEORY Basic Properties of Integers Modular Arithmetic Mathematical Induction
• GROUPS Symmetries of the Regular n-gon Introduction to Groups Properties of Group Elements Symmetric Groups Subgroups Lattice of Subgroups Group Homomorphisms Group Presentations Groups in Geometry Diffie-Hellman Public Key Semigroups and Monoids
• QUOTIENT GROUPS Cosets and Lagrange's Theorem Conjugacy and Normal Subgroups Quotient Groups Isomorphism Theorems Fundamental Theorem of Finitely Generated Abelian Groups
• RINGS Introduction to Rings Rings Generated by Elements Matrix Rings Ring Homomorphisms Ideals Quotient Rings Maximal and Prime Ideals
• DIVISIBILITY IN COMMUTATIVE RINGS Divisibility in Commutative Rings Rings of Fractions Euclidean Domains Unique Factorization Domains Factorization of Polynomials RSA Cryptography Algebraic Integers
• FIELD EXTENSIONS Introduction to Field Extensions Algebraic Extensions Solving Cubic and Quartic Equations Constructible Numbers Cyclotomic Extensions Splitting Fields and Algebraic Closures Finite Fields
• GROUP ACTIONS Introduction to Group Actions Orbits and Stabilizers Transitive Group Actions Groups Acting on Themselves Sylow's Theorem A Brief Introduction to Representations of Groups
• CLASSIFICATION OF GROUPS Composition Series and Solvable Groups Finite Simple Groups Semidirect Product. Classification Theorems Nilpotent Groups
• MODULES AND ALGEBRAS Boolean Algebras Vector Spaces Introduction to Modules Homomorphisms and Quotient Modules Free Modules and Module Decomposition Finitely Generated Modules over PIDs, I Finitely Generated Modules over PIDs, II Applications to Linear Transformations Jordan Canonical Form Applications of the Jordan Canonical Form A Brief Introduction to Path Algebras
• GALOIS THEORY Automorphisms of Field Extensions Fundamental Theorem of Galois Theory First Applications of Galois Theory Galois Groups of Cyclotomic Extensions Symmetries among Roots-- The Discriminant Computing Galois Groups of Polynomials Fields of Finite Characteristic Solvability by Radicals
• MULTIVARIABLE POLYNOMIAL RINGS Introduction to Noetherian Rings Multivariable Polynomial Rings and Affine Space The Nullstellensatz Polynomial Division-- Monomial Orders Grobner Bases Buchberger's Algorithm Applications of Grobner Bases A Brief Introduction to Algebraic Geometry
• CATEGORIES Introduction to Categories Functors
• APPENDICES LIST OF NOTATIONS BIBLIOGRAPHY INDEX
• Projects appear at the end of each chapter.
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Introduction to Abstract Algebra

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