# Question on Galois

1. Dec 4, 2009

### dabien

1. The problem statement, all variables and given/known data

Prove that if p is a prime then $$(\mathbb{Z}/p^2)^x \cong C_p$$ $$\times$$ $$C_{p-1} \cong C_{p(p-1)}$$ where $$C_n$$ denotes the cyclic group of order n.

Next part is to describe the Galois structure of the field of $$p^2$$ roots of unity, p is prime for the cases of a. p=5, b. p=7. How many intermediate fields are there in these cases?

2. Relevant equations

3. The attempt at a solution
My idea was to consider a homomorphism $$\mathbb{Z}/p^2 \rightarrow \mathbb{Z}/p$$ and observe that $$(\mathbb{Z}/p)^\times$$ is a cyclic group of order p-1. Then maybe use proof by induction that if p is an odd prime then $$(\mathbb{Z}/p^n)^\times$$ is cyclic for all n.