# 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 $EF$. Then:

1. 1.

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

2. 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