Isomorphism to certain Galois group and cyclic groups

  • #1
784
11

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!
 

Answers and Replies

  • #2
15,129
12,838

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.
 
  • Like
Likes PsychonautQQ

Related Threads on Isomorphism to certain Galois group and cyclic groups

  • Last Post
Replies
1
Views
2K
Replies
11
Views
7K
Replies
14
Views
1K
Replies
3
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
9
Views
6K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
4
Views
2K
Top