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You can define a homomorphism from Q(μp(p−1)) 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(p−1)-th root of unity in ˆK. Now you can do f(something in K).valuation()
.