These are notes from a first term abstract algebra course, an introduction to groups, rings, and fields. There is an emphasis on specific examples.

I hope to revise these notes at some point, and also get notes for additional topics in abstract algebra written up. There's a need for more motivation, and more pictures.

The first link in each item is to a Web page; the second is to a PDF (Adobe Acrobat) file. Use the PDF if you want to print it.

[July 29, 2022] I corrected typos and made some minor cosmetic changes to the notes on groups and rings.

- Groups

[PDF] - Examples of groups

[PDF] - Subgroups

[PDF] - Group homomorphisms and isomorphisms

[PDF] - Matrix groups

[PDF] - Divisibility

[PDF] - Greatest common divisors; the Euclidean and Extended Euclidean algorithms

[PDF] - Prime numbers

[PDF] - Modular arithmetic

[PDF] - Cyclic groups

[PDF] - The group of units in the integers mod n; Fermat's theorem; Wilson's theorem

[PDF] - Permutations

[PDF] - Direct products

[PDF] - The Structure Theorem for Finitely Generated Abelian Groups

[PDF](Examples only - the theorem is stated, but not proved) - Cosets and Lagrange's theorem

[PDF] - Normal subgroups and quotient groups

[PDF] - The Universal Property of the Quotient

[PDF]

- The First Isomorphism Theorem

[PDF] - Group maps from Z
_{m}to Z_{n}

[PDF] - Rings

[PDF] - Ring homomorphisms and isomorphisms

[PDF] - Ideals and subrings

[PDF] - Integral domains and fields

[PDF] - Polynomial rings; the Root and Factor Theorems; polynomial gcds

[PDF] - Quotient fields

[PDF] - Quotient rings

[PDF] - Quotient rings of polynomial rings

[PDF]