All my notes are based on the Math 113 course at UC Berkeley. The midterm and practice questions are also taken from the course.
Topics:
In no particular order. Extensive guides on: Introductory Ring Theory Half-Factorial Domains Vector Spaces in Abstract Algebra Abstract algebra for ML Extension Fields Factorizations in Integral domains Fields Group Actions Cryptography Group Isomorphisms and Homomorphisms
Proofs:
Taken from problems I found interesting or fundamental theorems. Homomorphism is Injective if and only if the kernel is trivial All Ideals in integers mod n Cayley’s Theorem (Important applications) Order of elements in Cyclic Groups Proof of Kronecker’s Theorem Prove there are no injective homomorphisms Why every prime element is irreducible in an integral domain.
Quick Guides:
All Ideals in integers mod n Normal function of euclidian integers Behaviors of Z modulo Organization of R polynomial x by classification of R
Review:
Completed solutions of given practice problems and midterms. Group Theory and Basics: Midterm 1 Practice Midterm 1 Ring and Field Theory: Practice Final