# Finding p-adic valuations in high degree cyclotomic fields

I'm looking at a cyclotomic field ${\bf Q}(\mu_{p(p-1)})$ for $p$ a prime around 50 and so this field has fairly large degree. In this field, $p$ has ramification index $p$ and has $p-1$ primes sitting above it.

I'm trying to compute the valuation of an element in this field at any of these primes above $p$. Using commands like "primes_above" won't seem to work as the computer just hangs presumably because this extensions degree is just too big.

Questions:

1) Is there another way to compute $p$-adic valuations in this field?

2) Locally, this is only a $p$-th degree extension of ${\bf Q}_p$. So I created a p-adic field by using pAdicField(p).ext(1+(x+1)+(x+1)^2+...+(x+1)^(p-1)) to create this local p-th degree extension of Q_p. However, I can't find any way to map my global elements in ${\bf Q}(\mu_{p(p-1)})$ to this local field. Any ideas on how to proceed along these lines?