# Calculating Galois Group of extension

• PsychonautQQ
In summary, the conversation discusses finding the degree, Galois group, and intermediate fields of an extension Q(c,b):Q where c is a primitive 3rd root of unity and b is the third real root of four. The degree of the extension is 6 and it is Galois since it is a splitting field over Q. The Galois group is suspected to be S_3 and <n> and A_3 are the only proper subgroups. However, there is some uncertainty about finding intermediate fields.
PsychonautQQ

## Homework Statement

Let c be a primitive 3rd root of unity and b be the third real root of four. Now consider the extension Q(c,b):Q. Find the degree of this extension, show that it is Galois, and calculate Gal(Q(c,b):Q) and then use the Galois group to calculate all intermediate fields.

## The Attempt at a Solution

The minimal polynomial of b over Q is x^3-4. The minimal polynomial of c over Q(b) is x^2+x+1. Therefore [Q(c,b):Q]=6.

Char(Q)=0 so the extension is separable. Q(c,b) is a splitting field for x^3-4, thus it is a finite splitting field over Q. Thus the extension Q(c,b):Q is Galois.

I am having a bit of trouble calculating the Galois group. I know there must be a map m such that m(c)=c^2 whilst fixing everything else, and there is a map n such that n(b)=cb whilst fixing everything else. Therefore all the roots of the minimal polynomial of b can be obtained by applying n to b, i.e. n*n(b)=n(cb)=(c^2)b whilst applying n to (c^2)b gives n((c^2)b)=b. The map m will take c to the other root of it's minimal polynomial in Q and then back again, m(c) = c^2 and m(c^2)=(c^2)^2=c^4=c.

So my first (naïve) thought was that this Galois group would be z3 x z2, although I suspected this would be incorrect because there is some overlapping of the maps (since m takes b to a root that is now effected by the map n). I showed that m*n(b+c+c^2) does not equal n*m(b+c+c^2) and thus this group is not abelian. I know the order of the group must be 6 because that is the degree of the extension (which is Galois) and I know that the group is non abelian.

Non abelian groups of order 6... S_3, D_3, maybe some SDP Z3xZ2. Perhaps the group must be S_3 because it contains a 3-cycle (the map m) and a transposition (the map n).

Anyway, that's where I'm at. I believe the Galois group will be S_3. if this turns out to be true, then I will then need to find all subgroups of S_3 and then go through the gritty business of using the subgroups of the Galois group to find the intermediate fields. Subgroups of S_3 will be A_3, which is all elements of S_3 that can be expressed as an even number of transpositions and A_3 will have order 3, thus it will be cyclic of order 3. S_3 also has a subgroup generated by the transposition that is one of the two generators for S_3, in this case the map n. Thus <n> will be a subgroup of S_3 of order 2. So I believe that <n> and A_3 are the only two proper subgroups of S_3.

I'll leave it at this for now and have somebody check my work, for all I know the Galois group isn't even S_3, and if it is I'm not sure if it only has 2 proper subgroups.

Maybe I missed the point somewhere, because I don't get intermediate fields. But why isn't ##c## in the splitting field of ##b##?

## 1. What is a Galois group?

A Galois group is a fundamental concept in abstract algebra that describes the symmetry or automorphisms of a field extension. It is named after the French mathematician Évariste Galois.

## 2. How is the Galois group of an extension calculated?

The Galois group of an extension is calculated by first finding the roots of the extension's defining polynomial. Then, the group elements are determined by mapping each root to another root through the automorphisms of the field extension. The resulting group is the Galois group.

## 3. Why is calculating the Galois group important?

The Galois group provides important information about the structure and properties of a field extension. It can help determine if a polynomial is solvable by radicals, provide insight into the roots of a polynomial, and aid in solving equations in terms of radicals.

## 4. Can the Galois group of an extension be calculated for any extension?

No, the Galois group can only be calculated for finite extensions that are Galois extensions. A Galois extension is a finite extension for which the Galois group is isomorphic to the automorphism group of the extension.

## 5. Are there any applications of the Galois group in other fields?

Yes, the Galois group has applications in various fields such as cryptography, coding theory, and geometry. It is also used in the study of elliptic curves and their applications in number theory and cryptography.

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