Math 500. Abstract Algebra I
Textbook: Hungerford, Algebra, 5th Edition, Springer-Verlag, New York Inc., 1997
- 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.
- 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.
- 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.
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Syllabus Revised 8/27/03