Semester 1:

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

Semester 2:

• 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