Abstract Algebra refers to a broad area of mathematical ideas concerning structure such as groups, rings, fields, lattices, and vector spaces. Number systems like N,Z, or Q have unique properties which allow equations to be stated and solved; new number systems can be constructed by "abstracting" away these properties to get new more complex number systems. These structures unify many disparate areas with a common language and lay the groundwork for such diverse areas as divisibility, gauge theory in physics, quintic equations, and the fundamental group in topology. Many books present groups and fields as proceeding from lists of axioms and then derive theorems. This presents a false impression that this is how the theory developed, when in fact it was the opposite. Mathematicians like Cayley, Galois, and Lagrange studied particular things like permutations or hypercomplex numbers and "abstracted" the properties from these studies.

**Prerequisites: **Some basic skills in solving equations and "high-school level" math is needed. E.g. Khan Academy's Algebra videos.

## Textbooks Edit

- Pinter, A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics)
- Sethuraman,
*Rings, Fields, and Vector Spaces: An Introduction to Abstract Algebra via Geometric Constructibility (Undergraduate Texts in Mathematics)* - Dummit and Foote,
*Abstract Algebra* - Artin,
*Algebra* - Beachy and Blair,
*Abstract Algebra* - Aluffi,
*Algebra: Chapter 0*(also introduces categories) - Chevalley,
*Fundamental Concepts of Algebra* - Birkhoff and Maclane,
*Algebra* - Lang,
*Undergraduate Algebra* - Rotman,
*Advanced Modern Algebra: 3rd Edition* - Atiyah and MacDonald,
*Introduction to Commutative Algebra* - Hungerford,
*Algebra*

## Online Edit

- Judson, Abstract Algebra: Theory and Applications
- Milne, Course Notes (the pdfs on Group Theory, Fields and Galois Theory)
- Conrad, Blurbs (a large collection of pdf files on various subtopics)
- Harvard Math E-222 video lectures
- John Eliott, Group Theory Playlist (also see Ring, Field, and Galois playlists)

## Miscellaneous Edit

- Cayley, On the theory of groups, as depending on the symbolic equation θ^n=1
- Galois, Œuvres mathématiques d'Évariste Galois (french)
- Jordan, Traité des substitutions et des équations algébriques (first expository paper on groups)
- Lagrange,
*Réflexions sur la résolution algébrique des équations*(french) - Lie, Theorie der transformationsgruppen (german, vol 1 only, vols 2 and 3 available by search)
- Feit and Thompson, Solvability of groups of odd order (first published proof of Burnside's conjecture)