# Isomorphism to certain Galois group and cyclic groups

• PsychonautQQ

## 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*

## 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!

## 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*
Can you clarify the roles of ##m## and ##q## here? Is is ##m\,\cdot\, q= p-1## or ##m=q##?

## Homework Equations

Which theorems have already been proven in this context? It looks like you may use the correspondence between Galois extensions and automorphism groups, but you haven't mentioned it.

## 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!
So it is actually a unique element ##g_i## depending on ##i##! Which theorem have you used here?

See, if you assume the availability of all those theorems, then there will be nothing left to prove.

• PsychonautQQ