proof that the compositum of a Galois extension and another extension is Galois

Proof.

The diagram of the situation of the theorem is:

\xymatrix@R1pc@C1pc{&&&\ar@{-}[llld]\ar@{-}[rdd]EF\\ \ar@{-}[rdd]E\\ &&&&\ar@{-}[llld]F\\ &\ar@{-}[d]E\cap F\\ &K}

To see that $EF/F$ is Galois, note that since $E/K$ is Galois, $E$ is a splitting field of a set of polynomials over $K$ ; clearly $E F$ is a splitting field of the same set of polynomials over $F$ . Also, if $f\in K[x]$ is separable over $K$ , then also $f$ is separable over $F$ . Thus $E F$ is normal and separable over $F$ , so is Galois. $E$ is obviously Galois over $E\cap F$ since $E\cap F\supset K$ .

Let $r$ be the restriction map

r:H=\operatorname{Gal}(EF/F)\to\operatorname{Gal}(E/K):\sigma\mapsto\sigma|_{E}

$r$ is clearly a group homomorphism, and since $E$ is normal over $K$ , $r$ is well-defined.

Claim $r$ is injective. For suppose $\sigma\in\operatorname{Gal}(EF/F)$ and $\sigma|_{E}$ is the identity. Then $\sigma$ is fixed on $F$ (since it is in $\operatorname{Gal}(EF/F)$ and on $E$ (since its restriction to $E$ is the identity), so is fixed on $E F$ and thus is itself the identity.

Now, the image of $r$ is a subgroup of $\operatorname{Gal}(E/K)$ with fixed field $L$ , and thus the image of $r$ is $\operatorname{Gal}(E/L)$ . Claim $E\cap F=L$ . $\subset$ is obvious: any element $x\in E\cap F$ is fixed by $\sigma|_{E}$ for each $\sigma\in H$ since $\sigma$ fixes $F$ . Thus $E\cap F\subset L$ . To see the reverse inclusion, choose $x\in L$ ; then $x$ is fixed by each $r(\sigma)$ for $\sigma\in H$ . But $x\in L\subset E$ , so that (as an element of $E$ ), $x$ is fixed by each $\sigma\in H$ . Thus $x\in F$ so that $x\in E\cap F$ .

Thus $L=E\cap F$ , and $r$ is then an isomorphism $\operatorname{Gal}(EF/F)\cong\operatorname{Gal}(E/E\cap F)$ . ∎

References

1 Morandi, P., Field and Galois Theory, Springer, 1996.

Title	proof that the compositum of a Galois extension and another extension is Galois
Canonical name	ProofThatTheCompositumOfAGaloisExtensionAndAnotherExtensionIsGalois
Date of creation	2013-03-22 18:41:58
Last modified on	2013-03-22 18:41:58
Owner	rm50 (10146)
Last modified by	rm50 (10146)
Numerical id	6
Author	rm50 (10146)
Entry type	Proof
Classification	msc 11R32
Classification	msc 12F99