Math 500. Abstract Algebra I

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 S_n.
   • 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's 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.

Graduate Guide homepage

Department of Mathematics University of Illinois at Urbana-Champaign 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Telephone: (217) 333-3350    Fax (217) 333-9576 office@math.uiuc.edu

Syllabus Revised 8/27/03