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Milne, J. S., Fields and Galois Theory,
Kea Books, Ann Arbor, MI,
2022. x+190 pp. ISBN: 979-8-218-07399-2.
Description: This book gives a concise exposition of the theory
of fields, including the Galois theory of field
extensions, the Galois theory of étale algebras, and
the theory of transcendental extensions. The first five chapters
treat the material covered in most courses in Galois theory while
the final four are more advanced.
The first two chapters are concerned with preliminaries on polynomials
and field extensions, and Chapter 3 proves the fundamental theorems
in the Galois theory of fields. Chapter 4 explains, with copious examples,
how to compute Galois groups, and Chapter 5 describes the many applications
of Galois theory.
In Chapter 6, a weak form of the Axiom of Choice is used to show that
all fields admit algebraic closures, and that any two are isomorphic. The last three chapters
extend Galois theory to infinite field extensions, to étale algebras
over fields, and to nonalgebraic extensions.
The approach to Galois theory in Chapter 3 is that of Emil Artin, and
in Chapter 8 it is that of Alexander Grothendieck.
This book originated as the notes for a first-year graduate course
taught at the University of Michigan, but they have since been revised
and expanded numerous times. The only prerequisites are an undergraduate
course in abstract algebra and some group theory. There are ninety-six
exercises, most with solutions.
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