nculwell.github.io

Linear Algebra

See the Linear Algebra section.

Algebra (aka Abstract Algebra, Modern Algebra)

First books

• Pinter, A Course in Abstract Algebra (2e Dover)

  Pinter gives you a smooth start in algebra from the very beginning. Each chapter gives a short exposition (typically 4-8 pages) followed by lots of problems. Often the problems fill more pages than the text. In each chapter, problems begin with simple applications of definitions to concrete examples, progress to relatively simple proofs that follow from the definitions and theorems in the text to more difficult problems and, in an extended series of exercises, a proof of Sylow’s Theorem.

• Herstein, Abstract Algebra (3e hardcover OOP, 3e paperback) or Topics in Algebra (2e paperback, 2e intl)

  Herstein has lots of great problems. Topics is considered somewhat more challenging than AA.

• Gallian, Contemporary Abstract Algebra (8e paperback, 7e, 6e)

• Shoup. A Computational Introduction to Number Theory and Algebra (FREE ONLINE; 2e at Amazon)

• Connell, Elements of Abstract and Linear Algebra (FREE ONLINE)

  I haven’t looked at this one at all yet, but it’s free: http://www.math.miami.edu/~ec/book/

Major textbooks

• Artin, Algebra (2e [expensive!], 2e intl, 2e intl @AbeBooks)
• Jacobson, Basic Algebra I (2e Dover)
• Jacobson, Basic Algebra II (2e Dover)
• Dummit and Foote, Algebra (3e intl, 3e [expensive!], 2e)
• Mac Lane and Birkhoff, Algebra 3e (3e)
• Lang, Algebra (3e)
• Hungerford, Algebra (Springer hardcover)

Other books

• Beachy. Abstract Algebra: Study Guide (1e) - Contains problems with solutions.

Free resources

• J. S. Milne, course notes: http://www.jmilne.org/math/CourseNotes/index.html
• Ash. Abstract Algebra: The Basic Graduate Year (with solutions) http://www.math.uiuc.edu/~r-ash/Algebra.html
• Ash. A Course in Algebraic Number Theory (with solutions) http://www.math.uiuc.edu/~r-ash/ANT.html

Group theory

• Carter. Visual Group Theory (1e)
• Rose. A Course on Group Theory (Dover) - Covers the basics.
• Rotman. An Introduction to the Theory of Groups (4e)
• Scott. Group Theory (Dover) - Theorem-proof.
• Robinson. A Course in the Theory of Groups (2e) - A solid second course in group theory.
• Humphreys. A Course in Group Theory (1e)
• Dixon. Problems in Group Theory (Dover)
• Isaacs. Character Theory of Finite Groups (Dover)
• Kurzweil, Stellmacher. The Theory of Finite Groups: An Introduction (2004)
• Wilson. The Finite Simple Groups (1e)
• Lyndon, Schupp. Combinatorial Group Theory (Dover)

Lie theory (Lie groups/algebras, representation theory, linear algebraic groups)

• Tapp. Matrix Groups for Undergraduates (2e, 1e)
• Fulton and Harris. Representation Theory: A First Course (1e)
• Humphreys. Introduction to Lie Algebras and Representation Theory (1e)
• Serre. Linear Representations of Finite Groups (1e)
• Procesi. Lie Groups: An Approach through Invariants and Representations (1e)
• Warner. Foundations of Differentiable Manifolds and Lie Groups (1e)
• Stillwell. Naive Lie Theory (1e)
• Knapp. Lie Groups: Beyond an Introduction (2e)
• Goodman and Wallach. Symmetry, Representations, and Invariants (1e)
• Pollatsek. Lie Groups: A Problem Oriented Introduction via Matrix Groups (1e)
• Jacobson. Lie Algebras (Dover)
• Borel. Linear Algebraic Groups (2e)
• Humphreys. Linear Algebraic Groups (1e)
• Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (2e)
• Chevalley. Theory of Lie Groups (PUP 1946)
• James and Liebeck. Representations and Characters of Groups (2e)

Commutative algebra

• Eisenbud, Commutative Algebra: with a View Toward Algebraic Geometry (Springer paperback)
• Atiyah and MacDonald, Introduction To Commutative Algebra (Paperback)
• Matsumura, tr. Reid. Commutative Ring Theory (1e)
• Miller and Sturmfels. Combinatorial Commutative Algebra (1e)
• Ene and Herzog. Grobner Bases in Commutative Algebra (1e)
• Zariski and Samuel. Commutative Algebra (Vol I, Vol II)

Noncommutative algebra

• Lam. A First Course in Noncommutative Rings (2e)
• Lam. Lectures on Modules and Rings (1e)
• Lam. Exercises in Modules and Rings (1e)
• Farb and Dennis. Noncommutative Algebra (1e)

Number theory

Introductions

• Ireland and Rosen. A Classical Introduction to Modern Number Theory (2e)

  Perhaps the most popular introduction to number theory.

• Hardy and Wright. An Introduction to the Theory of Numbers (6e/2008 updated by Heath-Brown and Silverman, 5e/1980, 4e/1960 at Archive.org)

  A classic introduction to number theory. One of the most popular books. The 6th edition has been updated with recent developments.

• Davenport. The Higher Arithmetic: An Introduction to the Theory of Numbers (CUP 8e/2008, Dover 7e/1983 at AbeBooks)

  Another classic introduction to number theory. It’s quite short.

• Dudley. Elementary Number Theory (Dover 2e)

• Jones and Jones. Elementary Number Theory (1e corr)

• Andrews. Number Theory (Dover 1e)

  A combinatorial approach to elementary number theory. Includes the theory of partitions.

• Stillwell. Elements of Number Theory https://smile.amazon.com/dp/0387955879

• Crawford. Introduction to Number Theory (At AOPS, At Amazon)

• Pommersheim, Marks, Flapan. Number Theory: A Lively Introduction with Proofs, Applications, and Stories (1e)

• Samuel, tr. Silberger. Algebraic Theory of Numbers (Dover)

  Another short, classic introduction to number theory. A bit old-fashioned. It emphases algebraic structures, as the title suggests.

Beyond

• Apostol. Introduction to Analytic Number Theory (1e/1976 corr 1998)
• Koblitz. A Course in Number Theory and Cryptography (2e)
• Koblitz. p-adic Numbers, p-adic Analysis, and Zeta-Functions (2e)
• Koblitz. Introduction to Elliptic Curves and Modular Forms (2e)
• Edwards. Riemann’s Zeta Function (Dover)
• Andrews. The Theory of Partitions (1e)
• Andrews and Eriksson. Integer Partitions (2e)

Ramanujan

Online copies of Ramanujan’s work can be found here: http://www.imsc.res.in/~rao/ramanujan/contentindex.html

• Berndt. Number Theory in the Spirit of Ramanujan (1e) - An introduction to Ramanujan’s work for someone already acquainted with number theory.
• Ramanujan, ed. Hardy, Aiyar, Wilson. Collected Papers of Srinivasa Ramanujan (1e)
• Hardy. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work (AMS Chelsea)
• Andrews and Berndt. Ramanujan’s Lost Notebook (Part I, etc.)