Suppose I have a field $k$ and two extensions $K/k$ and $L/k$ which are both abelian Galois extensions of $k$. Then (assuming $K$ and $L$ are both contained in some bigger field) is the compositum $KL$ an abelian Galois extension of $k$?
[Math] Compositum of abelian Galois extensions is also abelian
field-theorygalois-theorynumber theory
Best Answer
Let $\psi\colon \text{Gal}(KL/k) \rightarrow \text{Gal}(K/k)\times \text{Gal}(L/k)$ be a map such that $\psi(\sigma) = (\sigma|K, \sigma|L)$. $\psi$ is clearly a group homomorphism. It's easy to see that $\psi$ is injective. Hence $\text{Gal}(KL/k)$ is abelian.