T. Shaska: Abstract Algebra

MTH 4775-4776: Abstract Algebra

The course one of the toughest courses you will ever take as an undergraduate student. Make sure to take the course seriously
from the very beginning of the semester.

Textbook: Introduction to Algebra, T. Shaska

Description: A study of groups, rings, and fields. We will cover groups and rings during the first semester and fields during the
second semester.

Prerequisites: MTH 3002 with a grade of C or better.

Course policies: The course will be conducted in accordance to the Oakland University regulations and policies.
Details can be found here https://oakland.edu/provost/policies-and-procedures

Syllabus: For detailed syllabus click here Winter 2021

First Semester: MTH 4775

Preliminaries

Fundamentals

Algebraic operations
Congruences modulo $n$
Symmetries of a regular $n$-gon, dihedral groups
Permutations
Linear groups
Complex numbers and groups associated to them
The group of points in an algebraic curve

Groups

Basic properties of groups

Subgroups
Homomorphisms
Cyclic groups
Cosets and Lagrange's Theorem

Quotient Groups and Homomorphisms

Isomorphisms
Normal subgroups and factor groups
Isomorphism theorems
Cauchy's theorem
Conjugacy classes
Cayley's theorem

Groups acting on sets

Groups acting on sets
Some classical examples of group action
Symmetries
The modular group and the fundamental domain

Sylow theorem

Groups acting on themselves by conjugation
$p$-groups
Automorphisms of groups
Sylow theorems
Simple groups

Direct products and Abelian groups

Direct products
Finite Abelian groups
Free groups and Finitely generated Abelian groups
Canonical forms

Solvable Groups

Normal series and the Schreier theorem
Solvable groups
Nilpotent Groups

Rings

Rings

Introduction to rings
Ring homomorphisms and quotient rings
Ideals, nilradical, Jacobson's radical
Ring of fractions
Chinese remainder theorem

Euclidean rings, PID's, UFD's

Integral domains and fields
Euclidean domains
Principal ideal domains
Unique factorization domains

Polynomial rings

Polynomials
Polynomials over UFD's
Irreducibility of polynomials
Symmetric polynomials and discriminant
Formal power series

Local and Notherian rings

Introduction to local rings
Introduction to Notherian rings
Hilbert's basis theorem
Hilbert's basis theorem

Second Semester: MTH 4776

Theory of fields

Field theory

Introduction to fields
Field extensions
Finitely generated and finite extensions
Simple extensions
Finite fields

Algebraic Closure

Algebraic extensions revisited
Splitting fields
Normal extensions
Algebraic closure
Some classical problems

Galois theory

Automorphisms of fields
Separable Extensions
Galois extensions
Cyclotomic extensions
Norm and trace
Cyclic extensions
Fundamental theorem of Galois theory
Solvable extensions
Fundamental theorem of Algebra

Computing Galois groups of polynomials}

The Galois group of a polynomial
Galois groups of quartics
Galois groups of quintics
Determining the Galois group of higher degree polynomials
Polynomials with non-real roots

Abelian Extensions

Abelian extensions and Abelian closure
Roots of unity
Cyclotomic extensions
Cyclic Extensions
Kumer extensions
Artin-Schreier theory

Finite Fields

Basic definitions
Separable extensions
Constructing Finite Fields
Irreducibility of polynomials over finite fields
Artin-Schreier extensions
The algebraic closure of a finite field

Transcendental Extensions

Transcendental Extensions
L\"uroth and Castelnuovo theorem
Noether Normalization Lemma
Linearly disjoint extensions
Separable and Inseparable extensions

Tony Shaska