CAPUANO LAURA
(syllabus)
Fields extensions and their basic properties.
Algebraic closure of a field: existence and uniqueness. Kronecker's construction.
Splitting fields and normal extensions.
Separable, inseparable and purely inseparable extensions. Primitive element theorem.
Galois extensions. Galois group and Galois correspondence for finite extensions.
Prefinite groups and Krull topology. Galois correspondence for infinite extensions.
Galois group of an equation. Cyclotomic extensions. Generic equation of degree n.
Linear independence of characters. Trace and norm. Hilbert 90 theorem. Cyclic extensions and Kummer theory.
Solvable groups. Solvable and solvable by radicals extensions.
More examples and applications.
(reference books)
Algebra S. Bosch
Algebra S. Lang
Algebra M. Artin
Class Field Theory J. Neukirch
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Università Roma Tre