Isomorphism to certain Galois group and cyclic groups

In summary: So, make sure you state all the theorems you have used before proceeding.In summary, the conversation discusses the Galois group G(Q(c):Q) being isomorphic to Z_p*, where c is a pth root of unity and p is a prime number. It also mentions that if there is some m that divides p-1, then there is an extension K of Q for which G(K:Q) is isomorphic to Z_q*. The conversation also delves into the use of theorems, specifically the correspondence between Galois extensions and automorphism groups, and the existence of a unique element g_i in the Galois group of Q(c^m):Q. It is important to state all the theore
  • #1
PsychonautQQ
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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!
 
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  • #2
PsychonautQQ said:

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.
 
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Related to Isomorphism to certain Galois group and cyclic groups

1. What is isomorphism in relation to Galois groups and cyclic groups?

Isomorphism is a mathematical concept that describes a structural similarity between two groups. In the context of Galois groups and cyclic groups, isomorphism means that the two groups have the same underlying structure, although they may be presented differently.

2. How do you determine if two groups are isomorphic?

The easiest way to determine if two groups are isomorphic is to look for a bijective function between them. If a function exists that maps each element of one group to a unique element of the other group, and vice versa, then the two groups are isomorphic.

3. What is the significance of isomorphism in Galois theory?

Isomorphism is an important concept in Galois theory because it allows us to compare the structures of different Galois groups. By identifying isomorphic groups, we can understand the relationships between different fields and their automorphism groups.

4. Can a Galois group be isomorphic to a cyclic group?

Yes, a Galois group can be isomorphic to a cyclic group. In fact, every finite abelian group is isomorphic to a product of cyclic groups, so any Galois group that is abelian will be isomorphic to a product of cyclic groups.

5. How does isomorphism affect the solvability of equations?

Isomorphism can have a significant impact on the solvability of equations. In some cases, an isomorphic group may be easier to work with and may provide a solution to an unsolvable equation. Additionally, the isomorphism between two groups may reveal structural similarities that can be used to find a solution to an equation.

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