Finding p-adic valuations in high degree cyclotomic fields
I'm looking at a cyclotomic field Q(μp(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 Qp. 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 Q(μp(p−1)) to this local field. Any ideas on how to proceed along these lines?