Undergraduate Abstract Algebra

Semester I

1. Fundamentals

   1. Algebraic operations

   2. Congruences modulo $n$

   3. Symmetries of a regular $n$-gon, dihedral groups

   4. Permutations

   5. Linear groups

   6. Complex numbers and groups associated to them

2. Basic properties of groups

   2. 1. Subgroups

      2. Homomorphisms

      3. Cyclic groups

      4. Cosets and Lagrange's Theorem

3. Quotient Groups and Homomorphisms

   2. 1. Isomorphisms

      2. Normal subgroups and factor groups

      3. Isomorphism theorems

      4. Conjugacy classes

      5. Cayley's theorem

4. Groups acting on sets

   2. 1. Groups acting on sets

      2. Some classical examples of group action

      3. Symmetries

5. Sylow theorem

   2. 1. Groups acting on themselves by conjugation

      2. $p$-groups

      3. Automorphisms of groups

      4. Sylow theorems

      5. Simple groups

6. Direct products and Abelian groups

   2. 1. Direct products

      2. Finite Abelian groups

      3. Free groups and Finitely generated Abelian groups

7. Solvable Groups

   2. 1. Normal series and the Schreier theorem

      2. Solvable groups

      3. Nilpotent Groups

8. Extension and Cohomology (skip)

9. Rings

   1. Introduction to rings

   2. Ring homomorphisms and quotient rings

   3. Ideals, nilradical, Jacobson's radical

   4. Ring of fractions

   5. Chinese remainder theorem

10. Euclidean rings, PID's, UFD's

    1. Integral domains and fields

    2. Euclidean domains

    3. Principal ideal domains

    4. Unique factorization domains

11. Polynomial rings

    1. Polynomials

    2. Polynomials over UFD's

    3. Irreducibility of polynomials

    4. Symmetric polynomials and discriminant

Semester II

• 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