Abstract Algebra & Galois Theory

Studying Abstract Algebra & Galois Theory

Quotients are an organizing construction in algebra: they identify elements while preserving the operations that matter. Judson provides a broad route through groups, rings and fields with reading questions and exercises. Milne's field notes narrow the focus to extensions and Galois theory. Start with the broad text if well-defined maps or quotient rings remain uncertain; use the field route when you can already move between subgroups and homomorphisms.

What to work on

  • Construct quotient groups and use the first isomorphism theorem.
  • Distinguish prime and maximal ideals through quotient rings.
  • Compute degrees and bases of simple field extensions.
  • Match subgroups and intermediate fields in a finite Galois extension.

A starting text

Abstract Algebra: Theory and Applications — Thomas W. Judson; Sage exercises by Robert A. Beezer. For quotients read chapters 10, Normal Subgroups and Factor Groups, and 11, Homomorphisms. Review the ideal and quotient-ring material in chapter 16, Rings. For the field route use chapter 21, Fields, then sections 23.1-23.2 on field automorphisms and the fundamental theorem.

Choosing what to read

Build a quotient map and prove that its induced map is well-defined before using an isomorphism theorem. Compare ideals through their quotient rings, then compute a small field extension with an explicit basis. For the biquadratic extension of the rationals, draw the subgroup and intermediate-field correspondences and check each fixed field by hand. Separate normality and separability in the theorem's hypotheses, giving an example of what each condition controls.

Abstract Algebra: Theory and Applications

Thomas W. Judson; Sage exercises by Robert A. Beezer

Repair quotient constructions before attempting the Galois correspondence.

How it reads
Broad introductory algebra with separate group, ring and field routes.
What it expects
Proof-writing and linear algebra; earlier group material for quotient review.
What to keep in mind
Completing selected field sections is not a replacement for the earlier group and ring foundations.
Practice
Practice page for this work at https://judsonbooks.org/aata-files/aata-html/galois.html: Exercises provided; public solutions not verified. Chapter 23: section 23.4 Reading Questions and section 23.5 Exercises; use chapter 10 or 16 exercise sets when reviewing quotients.

Complete public HTML text.

Edition, reading range and source

For quotients read chapters 10, Normal Subgroups and Factor Groups, and 11, Homomorphisms. Review the ideal and quotient-ring material in chapter 16, Rings. For the field route use chapter 21, Fields, then sections 23.1-23.2 on field automorphisms and the fundamental theorem.

Source · Checked 2026-10-16

Fields and Galois Theory

J. S. Milne

Concentrate on field extensions after quotient and homomorphism arguments are secure.

How it reads
Focused course notes on extensions and Galois theory.
What it expects
Groups, rings and fields, as stated in the catalogue.
What to keep in mind
This is a field-theory continuation rather than a second introduction to all of abstract algebra.
Practice
Use the notes for field calculations and proof reconstruction; public solution coverage was not recorded for this item.

Complete author PDF, version 5.10, September 2022.

Edition, reading range and source

Version 5.10: chapters 1-3 for fields, splitting fields and the fundamental theorem of Galois theory; chapter 9 on transcendental extensions is a later continuation.

Source · Checked 2026-10-17

Practice

Galois-theory reading questions and exercises. Chapter 23: section 23.4 Reading Questions and section 23.5 Exercises; use chapter 10 or 16 exercise sets when reviewing quotients.. Exercises provided; public solutions not verified.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

  1. List the ideals of Z/12Z and identify which are prime or maximal by examining their quotient rings.

    A hint

    Pull ideals back to ideals of the integers containing (12). Include the zero ideal and the whole ring, but exclude the whole ring when testing primality.

    Work through the solution

    Write R=Z/12ZR=\mathbb Z/12\mathbb Z. By inverse image under the quotient map, its ideals correspond to (d)⊆Z(d)\subseteq\mathbb Z with (12)⊆(d)(12)\subseteq(d), equivalently positive divisors dd of 1212. They are (1ˉ),(2ˉ),(3ˉ),(4ˉ),(6ˉ),(12ˉ)(\bar1), (\bar2), (\bar3), (\bar4), (\bar6), (\bar{12}), the last being the zero ideal. For each such dd, R/(dˉ)≅Z/dZR/(\bar d)\cong\mathbb Z/d\mathbb Z.

    For d=2,3d=2,3 this quotient is a field, hence a domain, so these ideals are both maximal and prime. For d=4,6,12d=4,6,12 it has nonzero zero divisors: respectively 2⋅22\cdot2, 2⋅32\cdot3, and 3⋅43\cdot4 vanish modulo dd. Those ideals are neither prime nor maximal. The case d=1d=1 is the whole ring, excluded by the requirement that prime and maximal ideals be proper. This exhausts all ideals.

    If this was the sticking point:

Already comfortable with this? Skip quotient review if you can construct the first-isomorphism map and prove it is well-defined. Skip the field route if you can complete the biquadratic Galois correspondence and explain why separability and normality enter the general theorem.

Review the prerequisites · Compare the full subject collection

Where this can lead

Reading-source dates
  • Abstract Algebra: Theory and Applications · 2026-10-16
  • Galois-theory reading questions and exercises · 2026-10-16

Reading choices revised 2026-10-17.

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Group Theory

PDF

J. S. Milne · Course notes · Intermediate

A compact graduate introduction, including finite-group representations. Concise exposition makes prior exposure to algebra helpful.

Preparation: Undergraduate abstract algebra.

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Further print reading (2 retained references)

From the original catalogue. These are bibliographic references, not free full-text links; an authorized complete free edition is not currently linked here.

  • Algebra, 2nd edition — Michael Artin
  • Abstract Algebra, 3rd edition — David S. Dummit and Richard M. Foote

Additional reading

Search the complete catalogue.

Intermediate reading

Elements of Abstract and Linear Algebra

Edwin H. Connell · Textbook · Intermediate · English · Full-text access unverified

A terse course framework on groups, rings, matrix rings and vector spaces. Readers may need supplementary examples and should complete the omitted easy proofs.

Preparation: Proof-writing; vector spaces and linear maps.

Official page / access details
Edition and sources

Checked 2026-10-11

Algebra: Abstract and Concrete

Frederick M. Goodman · Textbook · Intermediate · English · Full-text access unverified

Develops groups, rings, fields, modules and Galois correspondence through examples. Assumes prior linear algebra.

Preparation: Proof-writing; vector spaces and linear maps.

Official page / access details
Edition and sources

Version: Edition 2.6, 2014, as listed by AIM · Checked 2026-10-11

Glimpses of Algebra and Geometry

Gábor Tóth · Textbook · Intermediate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Connects classical geometry and number-theoretic examples with algebra. A bridge between introductory courses and structural graduate algebra.

Preparation: Linear algebra and basic abstract algebra.

Official page / access details
Edition and sources

Version: Second edition, 2002 · Checked 2026-10-11

Undergraduate Algebra

Serge Lang · Textbook · Intermediate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A first structural algebra course with groups, rings, fields and matrix groups. This is distinct from the graduate Algebra reference.

Preparation: Linear algebra; proof-writing.

Official page / access details
Edition and sources

Version: Third edition, 2005 · Checked 2026-10-11

Abstract Algebra

Paul Garrett · Course notes · Intermediate · English · PDF

Complete 412-page PDF opened on the author's University of Minnesota site.

A full course moving from integers and polynomial arithmetic to groups, rings, fields and structural algebra.

Preparation: Comfort with elementary proofs and linear algebra.

Use: Use worked examples and exercises to move beyond computational algebra.

Open full PDFSource / terms
Edition and sources

Version: Author-hosted notes; date not stated · Checked 2026-10-13

Abstract Algebra: Tsinghua Course Notes

于品 (Pin Yu) · Course notes · Intermediate · Chinese · Online text

Online lecture text; chapters can be read separately.

A 2024 course built around groups, rings, modules, group actions and field extensions, with classical geometric questions as applications.

Preparation: Linear algebra; sets and maps; basic proof-writing.

Use: Read the definitions and group-action chapters together, working the concrete examples before the field-extension material.

Open online text / archive
Edition and sources

Original title: 抽象代数

Version: Fall 2024 course · Checked 2026-11-03

Lectures on Algebra

李文威 (Wen-Wei Li) · Author textbook · Intermediate · Chinese · PDF

Public full text; checked against the title and contents.

A first-year algebra course joining linear algebra with the language of groups, rings and modules. The public electronic edition is distinct from the later publication layout.

Preparation: School algebra and proof-writing.

Use: Use the vector-space and matrix chapters first, then follow the algebraic structures.

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Found via — References: 代数学讲义

Edition and sources

Original title: 代数学讲义

Version: 2026-10-16 electronic edition · Checked 2026-11-03

Graduate reading

Abstract Algebra

David S. Dummit and Richard M. Foote · Textbook · Graduate · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

A broad reference on groups, rings, modules, fields and Galois theory. Plan a selected route rather than treating its length as an entry requirement.

Preparation: Proof-writing; vector spaces and linear maps.

Official page / access details
Edition and sources

Version: Third edition, 2004 · Checked 2026-10-11

Algebra

Thomas W. Hungerford · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A broad graduate algebra text emphasizing groups, rings, modules and fields. Use selected parts after a first abstract-algebra course.

Preparation: Undergraduate algebra and proof-writing.

Official page / access details
Edition and sources

Version: 1974 · Checked 2026-10-11

Exercises in Classical Ring Theory

T. Y. Lam · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A problem-based companion for noncommutative ring theory: semisimple rings, radicals, division rings and related constructions.

Preparation: Rings, modules and a first graduate algebra course.

Official page / access details
Edition and sources

Version: Second edition, 2003 · Checked 2026-10-11

Lectures on the Algebraic Theory of Fields

K. G. Ramanathan · Long lecture notes · Graduate · English · PDF

Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.

A classical field-theoretic course for strengthening algebraic-extension and Galois arguments. Compare the historical presentation with a modern algebra text.

Preparation: Abstract algebra; fields and polynomial rings.

Open full PDFSource / terms
Edition and sources

Version: TIFR Lectures on Mathematics 5; authorized retypeset historical edition — original date on title page · Checked 2026-10-11

Algebra

Serge Lang · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A large structural algebra reference connecting groups, modules, fields and homological methods. Choose chapters around a course rather than reading every topic at once.

Preparation: Undergraduate abstract algebra.

Official page / access details
Edition and sources

Version: Revised third edition, 2002 · Checked 2026-10-11

A Course in Universal Algebra

Stanley Burris and H. P. Sankappanavar · Textbook · Graduate · English · Full-text access unverified

Authorized electronic edition; the link opens its download page.

Develops algebraic structures, varieties and logical connections beyond groups and rings. The author’s authorized download page includes revision information.

Preparation: Abstract algebra; elementary logic.

Official page / access details
Edition and sources

Version: Original 1981; author-hosted online editions · Checked 2026-10-11

Field Theory

Pete L. Clark · Long lecture notes · Graduate · English · PDF

Treats algebraic extensions and Galois theory, with further topics in field structure.

Preparation: Groups, rings and elementary field extensions.

Use: Extend a first algebra course into graduate field theory.

Open full PDFSource / terms
Edition and sources

Version: Author-hosted notes; date not stated · Checked 2026-10-13

Noncommutative Algebra

Pete L. Clark · Course notes · Graduate · English · PDF

Develops semisimple rings and modules, the Wedderburn–Artin theorem and radical methods.

Preparation: Rings, modules and group representations.

Use: Continue beyond commutative undergraduate ring theory.

Open full PDFSource / terms
Edition and sources

Version: 2011 summer lecture series; author-hosted notes · Checked 2026-10-13

An Invitation to General Algebra and Universal Constructions

George M. Bergman · Author manuscript · Graduate · English · PDF

Studies universal constructions, free objects and algebraic structures with categorical language and exercises.

Preparation: A first abstract algebra course and comfort with proofs.

Use: Develop structural algebra and universal-property reasoning.

Open full PDFSource / terms
Edition and sources

Version: Revised author manuscript, April 30, 2020 · Checked 2026-10-13

Basic Algebra

Anthony W. Knapp · Textbook · Graduate · English · PDF

Author provides the complete 764-page digital edition for noncommercial education, scholarship and research.

Connects linear algebra, group actions, rings, modules and Galois theory, with problem sets and extensive solution hints.

Preparation: Proof writing and first courses in linear algebra and abstract algebra.

Use: Consolidate undergraduate algebra and enter a graduate algebra sequence.

Open full PDFSource / terms
Edition and sources

Version: Digital second edition, 2016 · Checked 2026-10-13

Research reading

Lectures on Quadratic Jordan Algebras

Nathan Jacobson · Long lecture notes · Research · English · PDF

Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.

A specialist introduction to nonassociative algebra and quadratic Jordan structures. Follow an ordinary ring-and-module course first.

Preparation: Graduate algebra; nonassociative algebra foundations.

Open full PDFSource / terms
Edition and sources

Version: TIFR Lectures on Mathematics 45; authorized retypeset historical edition — original date on title page · Checked 2026-10-11

Chinese lecture notes

  • 抽象代数 — 于品 (Pin Yu). Read the definitions and group-action chapters together, working the concrete examples before the field-extension material.

  • 代数学讲义 — 李文威 (Wen-Wei Li). Use the vector-space and matrix chapters first, then follow the algebraic structures.