Math 401 Abstract Algebra I

Possible textbook: Hungerford, Algebra, 5th Edition, Springer-Verlag, New York Inc., 1997

1. Group theory
   • Isomorphism theorems and the factorization of homomorphisms using diagrams.
   • G-sets. Transitivity, orbits, stabilizers. Examples of actions of groups acting on coset spaces. Conjugacy classes. Normalizers and centralizers.
   • Symmetric groups. Alternating groups. Normal subgroups of Sn.
   • Subnormal and normal series. Schreier refinement and Jordan-Holder Theorem. Solvable groups.
   • Sylow's theorems.
   • Direct products of groups.
   • Examples of groups of small order, including dihedral and quaternion groups.
2. Commutative ring theory
   • Prime ideals. Maximal ideals. Examples of euclidean domains and principal ideal domains.
   • A PID is a UFD. Gauss' lemma. If R is a UFD then so is R[x]. Criteria for irreducibility of polynomials.
   • Zorn's lemma and applications to maximal ideals and bases of vector spaces.
3. Field theory
   • Algebraic and transcendental extensions. Existence and uniqueness of algebraic closures.
   • Separable and normal extensions. Extensions of automorphisms. Galois groups as permutation groups.
   • Splitting fields. Characterization of Galois extensions. Finite fields.
   • Fundamental theorem of Galois theory. Fundamental theorem of symmetric functions.
   • Examples of computations of Galois groups. Cyclic extensions.
   • Cyclotomic extensions. Irreducibility of the cyclotomic polynomial. Cyclicity of finite multiplicative groups in fields. Radical extensions.
   • Characterization of solvability by radicals.