ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 02 Sep 2015 16:45:41 -0500Finding p-adic valuations in high degree cyclotomic fieldshttp://ask.sagemath.org/question/26900/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?
Wed, 20 May 2015 08:47:43 -0500http://ask.sagemath.org/question/26900/finding-p-adic-valuations-in-high-degree-cyclotomic-fields/Answer by siggytm for <p>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. </p>
<p>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. </p>
<p>Questions:</p>
<p>1) Is there another way to compute $p$-adic valuations in this field?</p>
<p>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?</p>
http://ask.sagemath.org/question/26900/finding-p-adic-valuations-in-high-degree-cyclotomic-fields/?answer=29389#post-id-29389You can define a homomorphism from ${\bf Q}(\mu_{p(p-1)})$ to ${\bf Q}_p(\mu_p)$. 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 $p(p-1)$-th root of unity in $\hat K$. Now you can do `f(something in K).valuation()`.Wed, 02 Sep 2015 16:45:41 -0500http://ask.sagemath.org/question/26900/finding-p-adic-valuations-in-high-degree-cyclotomic-fields/?answer=29389#post-id-29389