PlanetMath: fundamental theorem of Galois theory

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fundamental theorem of Galois theory

(Theorem)

Let $L/F$ be a Galois extension of finite degree, with Galois group $G := \operatorname{Gal}(L/F)$ There is a bijective, inclusion-reversing correspondence between subgroups of $G$ and extensions of $F$ contained in $L$ given by

$K \mapsto \operatorname{Gal}(L/K)$ for any field $K$ with $F \subseteq K \subseteq L$
$H \mapsto L^H$ (the fixed field of $H$ in $L$ , for any subgroup $H \leq G$

The extension $L^H/F$ is normal if and only if $H$ is a normal subgroup of $G$ and in this case the homomorphism $G \longrightarrow \operatorname{Gal}(L^H/F)$ given by $\sigma \mapsto \sigma|_{L^H}$ induces (via the first isomorphism theorem) a natural identification $\operatorname{Gal}(L^H/F) = G/H$ between the Galois group of $L^H/F$ and the quotient group $G/H$

For the case of Galois extensions of infinite degree, see the entry on infinite Galois theory.

"fundamental theorem of Galois theory" is owned by yark. [ full author list (2) | owner history (1) ]

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Other names:

Galois theory, Galois correspondence

Attachments:: proof of fundamental theorem of Galois theory (Proof) by pbruin

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Cross-references: infinite Galois theory, quotient group, first isomorphism theorem, induces, homomorphism, normal subgroup, normal, fixed field, field, contained, extensions, subgroups, bijective, Galois group, degree, Galois extension
There are 32 references to this entry.

This is version 5 of fundamental theorem of Galois theory, born on 2002-01-05, modified 2007-07-04.
Object id is 1327, canonical name is FundamentalTheoremOfGaloisTheory.
Accessed 14973 times total.

Classification:

AMS MSC:	12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory)
	11R32 (Number theory :: Algebraic number theory: global fields :: Galois theory)
	11S20 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Galois theory)
	13B05 (Commutative rings and algebras :: Ring extensions and related topics :: Galois theory)

Pending Errata and Addenda

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