Let $K$ is a number field,and $H_{K}^{i},i=1,2,\cdots$ be its Hilbert class field tower,suppose it is finite,and let $L=H_{K}^{n}$ is the top of the tower. Must $L$ be galois over $K$?
$\begingroup$
$\endgroup$
3
-
7$\begingroup$ Each $H_K^i$ is Galois over $K$. Let $\sigma$ be any automorphism of the algebraic closure of $K$ fixing $K$. Inductively show that $\sigma(H_K^{i+1})$ is the maximal unramified abelian extension of $\sigma(H_K^i)=H_K^i$, so equals $H_K^{i+1}$. $\endgroup$– WojowuCommented Mar 20, 2020 at 15:23
-
$\begingroup$ Thank you very much ! $\endgroup$– fool rabbitCommented Mar 20, 2020 at 16:12
-
$\begingroup$ More generally (and basically for the same reason), if $L/K$ is Galois and $\mathfrak{m}$ is a Galois stable modulus of $L$, then the ray class field of $L$ with modulus $\mathfrak{m}$ is Galois over $K$. $\endgroup$– Alex B.Commented Mar 21, 2020 at 18:54
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
Posting as an answer to get out of the unanswered list.
Each field $H^i_K$ in the class field tower of $K$ is Galois over $K$. Indeed, let $\sigma$ be any automorphism of the algebraic closure of $K$ fixing $K$. Inductively we can show that $\sigma(H^i_K)=H^i_K$. Indeed, this is clear for $i=0$ (setting $H^0_K=K$) and assuming the equality for $i$, $\sigma(H^{i+1}_K)$ is the maximal unramified abelian extension of $\sigma(H^i_K)=H^i_K$, so equals $H^{i+1}_K$.
Therefore the union of the entire class field tower, be it finite or not, is Galois over $K$ as well.
