So there I have this problem:
Let $p$ be a prime number. Denote by E the algebraic closure of $\mathbb{Z}_{p}$. Prove that:
a) Every finite subfield F of E is a Galois extension of $\mathbb{Z}_{p}$.
b) For each positive integer n, there exists a subfield F of E such that the Galois group Gal(F,$\mathbb{Z}_{p}$) has order n and is also a solvable group.
For question a, I'm thinking of showing the normality and seperability of F. For b then I dont know how to start yet. Can I get any hint for the problem please?