the compositum of a Galois extension and another extension is Galois

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 $E F$ . Then:

1.

$E F$ is a Galois extension of $F$ and $E$ is Galois over $E\cap F$ ;
2.

Let $H=\operatorname{Gal}(EF/F)$ . The restriction map:

$\displaystyle H=\operatorname{Gal}(EF/F)$ $\displaystyle\longrightarrow$ $\displaystyle\operatorname{Gal}(E/E\cap F)$

$\displaystyle\sigma$ $\displaystyle\longrightarrow$ $\displaystyle\sigma|_{E}$

is an isomorphism, where $\sigma|_{E}$ 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.

Title	the compositum of a Galois extension and another extension is Galois
Canonical name	TheCompositumOfAGaloisExtensionAndAnotherExtensionIsGalois
Date of creation	2013-03-22 15:04:13
Last modified on	2013-03-22 15:04:13
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	6
Author	alozano (2414)
Entry type	Theorem
Classification	msc 12F99
Classification	msc 11R32
Related topic	FundamentalTheoremOfGaloisTheory
Related topic	GaloisExtension
Related topic	ExampleOfNormalExtension
Related topic	ClassNumberDivisibilityInExtensions
Related topic	GaloisGroupOfTheCompositumOfTwoGaloisExtensions
Related topic	ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility

	$\displaystyle H=\operatorname{Gal}(EF/F)$	$\displaystyle\longrightarrow$	$\displaystyle\operatorname{Gal}(E/E\cap F)$
	$\displaystyle\sigma$	$\displaystyle\longrightarrow$	$\displaystyle\sigma\|_{E}$