Galois Group of x^5 - 1, very confused

In summary, the conversation discusses finding the Galois group of x^5 - 1 over Q and identifying the subgroups and fixed subfields within it. The Galois group is found to be isomorphic to the multiplicative group of integers mod 5, which is in turn isomorphic to C4. The confusion arises when trying to find an intermediate field, but it is determined that an intermediate field does not need to contain any root. The intermediate field is then identified as Q(sqrt(5)). The conversation also briefly discusses whether the Galois group of a polynomial depends on the polynomial itself or just its splitting field.
  • #1
Zoe-b
98
0

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..
 
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  • #2
An intermediate field doesn't need to contain any root.
 
  • #3
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...?
 
  • #4
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!
 
  • #5
Zoe-b said:
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.

Zoe-b said:
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.
 
  • #6
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?
 
  • #7
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).
 
  • #8
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 :)
 

1. What is the Galois group of x^5 - 1?

The Galois group of x^5 - 1 is a mathematical concept related to field theory and polynomial equations. It is a group of automorphisms, or transformations, that preserve the roots of the polynomial x^5 - 1. In other words, the Galois group is a set of symmetries that can be applied to the roots of the polynomial without changing its overall structure.

2. How is the Galois group of x^5 - 1 calculated?

The Galois group of x^5 - 1 can be calculated using Galois theory, a branch of abstract algebra. This involves analyzing the structure of the polynomial and its roots, and determining which symmetries can be applied to the roots while still keeping the polynomial unchanged. The number of symmetries that can be applied determines the size and composition of the Galois group.

3. What are the practical applications of studying the Galois group of x^5 - 1?

The Galois group of x^5 - 1 has practical applications in fields such as cryptography, coding theory, and algebraic geometry. By understanding the symmetries of this polynomial, mathematicians can gain insight into the structure and properties of other mathematical objects. Additionally, the study of Galois groups has led to important advancements in abstract algebra and number theory.

4. What does it mean for the Galois group of x^5 - 1 to be "very confused"?

The term "very confused" is not a technical term used in mathematics. It may refer to the complexity of the Galois group of x^5 - 1, which can be difficult to understand and calculate. It may also indicate that the person asking the question is struggling to understand the concept or to find a solution to a particular problem related to the Galois group.

5. How does the Galois group of x^5 - 1 relate to other mathematical concepts?

The Galois group of x^5 - 1 is related to other mathematical concepts such as field extensions, group theory, and symmetry. It also has connections to other areas of mathematics such as algebraic geometry, topology, and number theory. Understanding the Galois group of x^5 - 1 can provide valuable insights into these related concepts and their applications.

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