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Finding p-adic valuations in high degree cyclotomic fields

asked 9 years ago

Robert Pollack gravatar image

I'm looking at a cyclotomic field Q(μp(p1)) for p a prime around 50 and so this field has fairly large degree. In this field, p has ramification index p and has p1 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(p1)) to this local field. Any ideas on how to proceed along these lines?

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answered 9 years ago

siggytm gravatar image

You can define a homomorphism from Q(μp(p1)) to Qp(μp). Here's an example, using p=47.

p=47
K.<zeta> = CyclotomicField(p*(p-1))  # this is the field you're interested in.
Khat.<pi> = Qp(p).ext(sum((1+x)^i for i in range(p)))  # its completion
a = Khat.residue_field().primitive_element()  # this is a primitive (p-1)-th root of unity in the residue field

In the example case with p=47, we get a=5.

zeta1 = Khat.teichmuller(ZZ(a))  # this is a primitive (p-1)-th root of unity in Khat.

I don't know why you need ZZ(a) instead of just a. I get 5 + O(pi) when I do Khat.teichmuller(a).

zeta2 = 1+pi  # this is a primitive p-th root
f = K.hom([zeta1*zeta2])

This f will be the homomorphism that sends the generator of K, above called zeta, to zeta1*zeta2, which is a primitive p(p1)-th root of unity in ˆK. Now you can do f(something in K).valuation().

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Asked: 9 years ago

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Last updated: Sep 02 '15