proof that the compositum of a Galois extension and another extension is Galois
Proof.
The diagram of the situation of the theorem is:
$$\text{xymatrix}\mathrm{@}R1pc\mathrm{@}C1pc\mathrm{\&}\mathrm{\&}\mathrm{\&}\text{ar}\mathrm{@}-[llld]\text{ar}\mathrm{@}-[rdd]EF\text{ar}\mathrm{@}-[rdd]E\mathrm{\&}\mathrm{\&}\mathrm{\&}\mathrm{\&}\text{ar}\mathrm{@}-[llld]F\mathrm{\&}\text{ar}\mathrm{@}-[d]E\cap F\mathrm{\&}K$$ |
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 $EF$ 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 $EF$ 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
$$r:H=\mathrm{Gal}(EF/F)\to \mathrm{Gal}(E/K):\sigma \mapsto {\sigma |}_{E}$$ |
$r$ is clearly a group homomorphism^{}, and since $E$ is normal over $K$, $r$ is well-defined.
Claim $r$ is injective^{}. For suppose $\sigma \in \mathrm{Gal}(EF/F)$ and ${\sigma |}_{E}$ is the identity^{}. Then $\sigma $ is fixed on $F$ (since it is in $\mathrm{Gal}(EF/F)$ and on $E$ (since its restriction to $E$ is the identity), so is fixed on $EF$ and thus is itself the identity.
Now, the image of $r$ is a subgroup^{} of $\mathrm{Gal}(E/K)$ with fixed field $L$, and thus the image of $r$ is $\mathrm{Gal}(E/L)$. 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^{} $\mathrm{Gal}(EF/F)\cong \mathrm{Gal}(E/E\cap F)$. ∎
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 |