Milne J.S. Fields and Galois Theory

Web Publication (v. 4.22). — [v2.01 (August 21, 1996). First version on the web.], 2011. — 126 p., eBook, English.[Single paper copies for noncommercial personal use may be made without explicit permission
from the copyright holder].
These notes give a concise exposition of the theory of fields, including the Galois theory of finite and infinite extensions and the theory of transcendental extensions.Notations.Basic Definitions and Results.
Rings.
Fields.
The characteristic of a field.
Review of polynomial rings.
Factoring polynomials.
Extension fields.
Construction of some extension fields.
Stem fields.
The subring generated by a subset.
The subfield generated by a subset.
Algebraic and transcendental elements.
Transcendental numbers.
Constructions with straight-edge and compass.
Algebraically closed fields.
Exercises.
Splitting Fields; Multiple Roots.
Maps from simple extensions.
Splitting fields.
Multiple roots.
Exercises.
The Fundamental Theorem of Galois Theory.
Groups of automorphisms of fields.
Separable, normal, and Galois extensions.
The fundamental theorem of Galois theory.
Examples.
Constructible numbers revisited.
The Galois group of a polynomial.
Solvability of equations.
Exercises.
Computing Galois Groups.
When is Gf An?
When is Gf transitive?
Polynomials of degree at most three.
Quartic polynomials.
Examples of polynomials with Sp as Galois group over Q.
Finite fields.
Computing Galois groups over Q.
Exercises.
Applications of Galois Theory.
Primitive element theorem.
Fundamental Theorem of Algebra.
Cyclotomic extensions.
Dedekind’s theorem on the independence of characters.
The normal basis theorem.
Hilbert’s Theorem 90.
Cyclic extensions.
Kummer theory.
Proof of Galois’s solvability theorem.
The general polynomial of degree n.
Norms and traces.
Exercises.
Algebraic Closures.
Zorn’s lemma.
First proof of the existence of algebraic closures.
Second proof of the existence of algebraic closures.
Third proof of the existence of algebraic closures.
(Non) uniqueness of algebraic closures.
Separable closures.
Infinite Galois Extensions.
Topological groups.
The Krull topology on the Galois group.
The fundamental theorem of infinite Galois theory.
Galois groups as inverse limits.
Nonopen subgroups of finite index.
Transcendental Extensions.
Algebraic independence.
Transcendence bases.
Luroth’s theorem. ¨
Separating transcendence bases.
Transcendental Galois theory.
A Review Exercises.
B Two-hour Examination.
C Solutions to the Exercises.