Field Arithmetic
UGA MATH 8430, Spring 2017.
Field extensions, Galois theory and inverse Galois theory, field invariants, Galois cohomology, and open problems.
dkrashen@uga.edu
Preliminary Syllabus (topic wishlist) for MATH8430
(Field Arithmetic)
- Finite field extensions and Galois theory
- Inseparable extensions, p-bases
- Kummer theory and Artin-Schrier theory, Hilbert 90
- Orderings and real closed fields
- Transcendental extensions, transcendence bases, regular extensions
- The Luroth problem
- Tsen-Lang theory, Diophantine dimension, Leep’s An property
- Other field invariants: pythagoras number, level, u-invariant
- Valuations, places, Witt vectors
- Derivations, Galois theory of simple purely inseparable extenions
- Tensor products, composita
- Tensors, algebraic structures, descent, the group H1
- Galois extensions of rings, generic Galois theory
- Inverse Galois theory, extension problems, the group H2
- Infinite Galois theory, profinite groups
- Connections to geometry and the Anabelian conjectures
- Galois cohomology and cohomological invariants of algebraic structures
- Milnor K-theory, reciprocity and ramification
- The Galois cohomology ring and Bloch-Kato conjectures
- Cup products, cohomological dimension and the cyclicity problem
- Witt rings and the Milnor conjectures
- The symbol length and period-index problems
- Motivic complexes