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## Homework Statement

Let c be a pth root of unit where p is prime. Then the Galois group G(Q(c):Q) is isomorphic to Z_p*. Show that if there is some m that divides p-1, then there is an extension K of Q such that G(K:Q) is isomorphic to Z_q*

## Homework Equations

## The Attempt at a Solution

I suspect that K=Q(c^m) is the field extension that we are looking for.

Then the basis elements for Q(c^m):Q are {1,c^m,c^2m,.....,c^(r)m) where r+1 is the order of <c^m> a subgroup of <c>. In the Galois group of Q(c^m):Q there will exist a unique element g such that g(1)=c^(im)

for each 0,........,r possible exponents of the basis elements.

Am I on the right track here? Thanks!