Suppose K is a field containing p th roots of unity. Suppose $[L:K]$ is a Galois extention with Galois group $G$. Let $ a \in L$ not in $K$ s.t $a^p \in K$. Describe a nontrivial map $G \to \Bbb Z/p\Bbb Z$.
I am thinking about $f(a)=a^2$ such that $f|_K=id$. Does this define $f$ completely? And after that does $G \to \Bbb Z/p\Bbb Z$ map would be $f \to 2$. I think my answer is meaningless.
- The first step should be proving $L=K(a)$ or something and the considering
- a map from $G \to \Bbb Z/\Bbb Z_p$.
Please help me with a full answer. I think I have solved this before but now I can't out of panic. I have an exam on Monday so any help will be appreciated.
Here are some hints to get you started.
Let $b = a^p$, so that $a$ is some choice of $\sqrt[p]b$. In particular, the field $K(\sqrt[p]b)$ is a subfield of $L$.
Question 1: What is the Galois group of $K(\sqrt[p]b)/K$? Here, you will need to use the fact that $\zeta_p\in K$ - otherwise this extension may not be Galois. You will also need to use the fact that $p$ is prime and that $a\notin K$ - otherwise this extension might not have the right degree.
Question 2: Can you describe a map $\mathrm{Gal}(L/K)\to\mathrm{Gal}(K(\sqrt[p]b)/K)$? Why is this map a surjective group homomorphism?