Galois Theory
This chapter contains the core of Galois theory. We study the group of automorphisms of a finite (and sometimes infinite) Galois extension at length, and give examples, such as cyclotomic extensions, abelian extensions, and even non-abelian ones, leading
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VI
Galois Theory
This chapter contains the core of Galoi s theory . We study the group of automorphisms of a finite (and sometimes infinite) Galois extension at length , and give examples , such as cyclotomic extensions, abelian extensions, and even non-abelian ones, leading into the study of matrix representations of the Galois group and their cla ssifications . We shall mention a number of fundamental unsolved problems , the most notable of which is whether given a finite group G , there exists a Galois extension of Q having this group as Galois group . Three surveys give recent points of view on those question s and sizeable bibliographies: B. MATZAT, Konstruktive Galoistheorie, Springer Lecture Notes 1284, 1987 B. MATZAT , Uber das Umkehrproblem der Galoisschen Theorie, lahrsbericht Deutsch . Mat .-Verein. 90 (1988), pp. 155-183 J. P.
S ERRE,
Topics in Galois theory, course at Harvard, 1989 , Jones and Bartlett,
Boston 1992 More specific references will be given in the text at the appropriate moment concerning this problem and the problem of determining Galois groups over spec ific fields, especially the rational numbers .
§1.
GALOIS EXTENSIONS
Let K be a field and let G be a group of automorphisms of K . We denote by KG the subset of K consisting of all elements x E K such that x" = x for all (J E G. It is also called the fixed field of G. It is a field because if x, y E KG then (x
+ y)a =
x"
+ ya =
X
+Y 261
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GALOIS THEORY
VI, §1
for all (J E G, and similarly, on e verifies that K is closed under multiplication, subtraction, and multiplicative inverse. Furthermore, KG contains 0 and 1, hence contains the prime field. An algebraic extension K of a field k is called Galois if it is normal and separable. We consider K as embedded in an algebraic closure. The group of automorphisms of Kover k is called the Galois group of Kover k, and is denoted by G(K/k), GK /b Gal(K/k), or simply G. It coincides with the set of embeddings of Kin 10 over k. For the convenience of the reader, we shall now state the main result of the Galois theory for finite Galois extensions.
Theorem 1.1. Let K be a finite Galois extension of k, with Galois group G. There is a bijection between the set of subfields E of K containing k, and the set of subgroups H ofG, given by E = K H • Thefield E is Galois over k if and only if H is normal in G, and if that is the case, then the map (J H (J IE induces an isomorphism ofGIH onto the Galois group of E over k. We shall give the proofs step by step, and as far as possible, we give them for infinite extensions.
Theorem 1.2. Let K be a Galois extension of k. Let G be its Galois group. Then k = KG. If F is an intermediate field, k cz F c K, then K is Galois over F. The map FH G(KIF) from the set of intermediate fields into the set of subgroups of G is injective. Proof Let o: E KG. Let (J be any embedding of k(r:x.) in K", inducing the identity on k. Extend (J to an embedding of K into K", and call this extension (J also.
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