What Is the Galois Group of x^5 - 1 Over Q?

[Zoe-b]

Homework Statement

I'm trying to find the galois group of x^5 - 1 over Q, and then for each subgroup of the galois group identify which subfield is fixed.

Homework Equations

The Attempt at a Solution

If w = exp(2*I*PI/5), then the roots not in Q are w, w^2, w^3, w^4. Its fairly easy to see by checking possible automorphisms that the Galois group is isomorphic to the multiplicative group of integers mod 5, which is in turn isomorphic to C4.

There may be a mistake there already.. but I'm not sure.

Here comes the confusion- C4 has one non-trivial proper subgroup, in this case found by multiplying only the elements w and w^4 together. But there are no intermediate fields as far as I can tell, since each root is primitive and so generates the other roots.

Please help! I've been teaching myself this course so I think I've probably just misunderstood something somewhere but not sure what..

 

[Hurkyl]

An intermediate field doesn't need to contain any root.

 

[Zoe-b]

Ok... true. Will an intermediate field be one over which x^4 + x^3 + x^2 + x + 1 splits into two quadratics? or is the polynomial irrelevant for this...?

 

[Zoe-b]

Think I've got it now- the intermediate field is Q(sqrt(5)) which is fixed by the subgroup {e,t} where e is the identity and t sends w to w^4, w^2 to w^3, that is, t is equivalent to complex conjugation.

Thank you!

 

[Hurkyl]

Think I've got it now- the intermediate field is Q(sqrt(5)) which is fixed by the subgroup {e,t} where e is the identity and t sends w to w^4, w^2 to w^3, that is, t is equivalent to complex conjugation.

Thank you!

This sounds right.

Ok... true. Will an intermediate field be one over which x^4 + x^3 + x^2 + x + 1 splits into two quadratics? or is the polynomial irrelevant for this...?

I expect this to be true as well.

 

[Zoe-b]

Fantastic- I have a more general question which as yet I've been unable to find the answer to in a textbook..
Does the galois group of a polynomial depend purely on its splitting field? Or is it in some way connected to the polynomial itself? For example, if two polynomials have different roots but the same splitting field, are their galois groups the same?

 

[Hurkyl]

The definition I know of the Galois group of a polynomial is literally that it is the Galois group of the splitting field (over the relevant base field).

 

[Zoe-b]

Ok thank you I thought that was the case but then got confused by questions where the splitting field seemed to be the same for different examples :)