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[parent] the compositum of a Galois extension and another extension is Galois (Theorem)
Theorem 1   Let $ E/K$ be a Galois extension of fields, let $ F/K$ be an arbitrary extension and assume that $ E$ and $ F$ are both subfields of some other larger field $ T$. The compositum of $ E$ and $ F$ is here denoted by $ EF$. Then:
  1. $ EF$ is a Galois extension of $ F$ and $ E$ is Galois over $ E\cap F$;
  2. Let $ H=\operatorname{Gal}(EF/F)$. The restriction map:
    $\displaystyle H=\operatorname{Gal}(EF/F)$ $\displaystyle \longrightarrow$ $\displaystyle \operatorname{Gal}(E/E\cap F)$  
    $\displaystyle \sigma$ $\displaystyle \longrightarrow$ $\displaystyle \sigma \vert _{E}$  

    is an isomorphism, where $ \sigma \vert _{E}$ denotes the restriction of $ \sigma$ to $ E$.
Remark 1   Notice, however, that if $ E/F$ and $ F/K$ are both Galois extensions, the extension $ E/K$ need not be Galois. See example of normal extension for a counterexample.



"the compositum of a Galois extension and another extension is Galois" is owned by alozano.
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See Also: fundamental theorem of Galois theory, Galois extension, example of normal extension, class number divisibility in extensions, Galois group of the compositum of two Galois extensions, extensions without unramified subextensions and class number divisibility

Keywords:  compositum, composite field, Galois extension

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Cross-references: counterexample, example of normal extension, isomorphism, map, restriction, compositum, subfields, extension, fields, Galois extension
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This is version 3 of the compositum of a Galois extension and another extension is Galois, born on 2005-02-21, modified 2005-03-10.
Object id is 6791, canonical name is CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois.
Accessed 1310 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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