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You 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().