The second term
of Math 120 will be devoted primarily to Field and Galois Theory. Time
permitting, topics should include:
Field extensions: Algebraic closure. Splitting fields and normal
extensions. Separable and Inseparable extensions and degree. Primitive
element theorem. Linear independence of characters. Traces and norms.
Normal basis theorem.
Galois theory: Galois
extensions. Absolute Galois group. Galois theory for infinite
extensions.
Examples: Finite fields.
Cyclotomic fields.
Galois
Cohomology: Galois Cohomology. Hilbert's 90 theorem.
Kummer
Theory: Cyclic extensions. Solvable extensions. Kummer Theory.
Transcendental
Extensions:
Transcendental extensions. Transcendence bases
theorem.
Ring extensions: Integral
extensions. Noether Normalization Theorem. Integral closure. Integral
Galois extensions. Going-up theorem. Decomposition groups. Extension
of homomorphisms. Valuation rings.
Complements: Introduction
to local fields. Dedekind domains. DVR's.
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