TITLE: A problem in Galois Theory
QUESTION [5 upvotes]: While reading algebraic number theory, I came across the following statement:
Let $K$ be a galois extension over $\mathbb{Q}$ and $H$ be the Hilbert class field (maximal unramified abelian extension) of $K$. As $H$ is maximal, $H$ is also galois over $\mathbb{Q}$. Can anyone explain why maximality implies $H$ is a galois extension over $\mathbb{Q}$?

REPLY [4 votes]: Sorry, my comment had a typo which made it unhelpful (the $K$ should be an $H$).
Here is the structure of the argument (with some details left to be filled in).  Suppose $H/\mathbb{Q}$ were not Galois.  Then there exists a conjugate $H^\sigma$ of $H$ distinct from $H$, and $H^\sigma$ is an unramified abelian extension of $K^\sigma$.  Since $K/\mathbb{Q}$ is Galois, $K^\sigma=K$, and so both $H$ and $H^\sigma$ are both unramified abelian extensions of $K$.  Thus their compositum $HH^\sigma$ is as well, and by assumed distinctness, properly contains $H$.  This contradicts that $H$ was the maximal unramified abelian extension of $K$.