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[parent] Galois group of the compositum of two Galois extensions (Theorem)
Theorem 1   Let $ E$ and $ F$ be Galois extensions of a field $ K$. Then:
  1. The intersection $ E\cap F$ is Galois over $ K$.
  2. The compositum $ EF$ is Galois over $ K$. Moreover, the Galois group $ \operatorname{Gal}(EF/K)$ is isomorphic to the subgroup $ H$ of the direct product $ G=\operatorname{Gal}(E/K)\times \operatorname{Gal}(F/K)$ given by:
    $\displaystyle H=\{ (\sigma, \psi) : \sigma\vert _{E\cap F}=\psi\vert _{E\cap F} \}$
    i. e. $ H$ consists of pairs of elements of $ G$ whose restrictions to $ E\cap F$ are equal.
Corollary 1   Let $ E$ and $ F$ be Galois extensions of a field $ K$ such that $ E\cap F=K$. Then $ EF$ is Galois over $ K$ and the Galois group is isomorphic to the direct product:
$\displaystyle \operatorname{Gal}(EF/K)\cong \operatorname{Gal}(E/K) \times \operatorname{Gal}(F/K).$



"Galois group of the compositum of two Galois extensions" is owned by alozano.
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See Also: the compositum of a Galois extension and another extension is Galois, Galois extension

Keywords:  compositum, Galois group

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Cross-references: restrictions, direct product, subgroup, isomorphic, Galois group, compositum, intersection, field, Galois extensions
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This is version 2 of Galois group of the compositum of two Galois extensions, born on 2005-02-22, modified 2005-03-10.
Object id is 6794, canonical name is GaloisGroupOfTheCompositumOfTwoGaloisExtensions.
Accessed 1483 times total.

Classification:
AMS MSC: 12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous)
 11R32 (Number theory :: Algebraic number theory: global fields :: Galois theory)
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