# (Galois Theory) Let $E/K$ be a Galois extension with Galois group isomorphic to $Z_{12}$. Determine the subfield lattice for $E/K$

(Galois Theory) Let $E/K$ be a Galois extension with Galois group isomorphic to $Z_{12}$. Determine the subfield lattice for $E/K$.

This is my rough proof to this question. I was wondering if anybody can look over it and see if I made a mistake or if there is a simpler way of doing this problem. So lets begin:

$E/K$ is Galois extension with Galois group isomorphic to $Z_{12}$ we need to determine subfield of lattice for $Z_{12}$

We know that subfield of $Z_{12}$ will correspond to positive divisors of $12$, which are $(1,2,3,4,6,12)$.

Hence subgroups of $Z_{12}$ would be $\langle 0\rangle$, $\langle6\rangle$, $\langle4\rangle$, $\langle3\rangle$, $\langle2\rangle$, $\langle1\rangle$.

(Because $Z_{12}=\langle0\rangle$)

Subfield of lattice would be:

I want thank you ahead of time for taking the time to look at this problem.

• Looks good to me. – Matt Samuel May 5 '15 at 19:23
• except that I guess you mean $Z_{12}=\langle 1\rangle$. – Hagen von Eitzen May 5 '15 at 19:38
• Looks like you drew the sub-group lattice not the sub-field lattice. When you translate into fields, make sure to flip it over, with $E$ at the top and $K$ at the bottom. – Gregory Grant May 5 '15 at 19:42