Galois group

The Galois group (after Évariste Galois ) is a group, with the help of field extensions in algebra can be studied.

The intermediate body of a field extension can be assigned to certain subgroups of the Galois group. So you can bring structural studies of field extensions with group theoretical investigations in connection. As to finite field extensions are finite Galois groups, so that such structural studies can often be greatly simplified.

Was historically significant that the classic questions of constructability - with ruler and compass - certain algebraic numbers so that could be translated into a group-theoretical formulation. For details about the classic question of constructibility by ruler and compass, examples and their modern solution is added to konstruierbares polygon.

• 3.1 seclusion

• 4.1 finite Körpererweitung
• 4.2 infinite-dimensional algebraic extension

• 5.1 Galois group of a cubic polynomial

Definition

Let be a field extension (read " body extension over "). That is, and are the body and the body K is included as a sub- ring in F. will thus become a (not necessarily finite-dimensional ) vector space.

In this situation, the group of all Körperautomorphismen the extension field F, the firm let the ground field K element-wise, the Galois group of F over K is called. formalized:

This can also be formulated as follows: the Galois group of F over K consists exactly of the Körperautomorphismen of F, which are also Vektorraumendomorphismen of F as a K- vector space.

Galois group of a polynomial

Let K be a body. As the Galois group of the polynomial f in the polynomial, the group is called, where F is a splitting field of the polynomial f. One speaks in this case of the splitting field since splitting field - and hence the Galois group of a polynomial - are uniquely determined up to isomorphism.

The splitting field of a polynomial F is normal to the body K. In this case, that is - nite here - body extension already then Galois if the irreducible factors of f over K is separable. The article deals with the notion of Galois theory Galois group of a polynomial, in this case satisfies the below mentioned first version of the main theorem - the main theorem for finite Galois extensions.

Different meanings of the term

Particularly useful is the Galois group if the field extension is a Galois extension (see below). In the literature it is often spoken of only in this case of " Galois group ". Then the group used in this Article, the K - automorphisms of F is called.

Properties

• The Galois group is a subgroup of the automorphism group of F.
• If the field extension finite, i.e., is finite-dimensional over, so is the group of order less than or equal to the degree of expansion. In this case, there is the minimal polynomial of over for each body part.
• Let F be a splitting field of the polynomial f over K. Every automorphism of the Galois group of the polynomial f represents a zero of back to a zero. So the Galois group acts on the set of zeros of the body, as a permutation group and is thus isomorphic to a subgroup of the symmetric group. A separables through irreducible polynomial that operation is even transitive, that is, at two different sites, there is a zero element of the Galois that maps to.

Galoiskorrespondenz, Closed subgroups and intermediate body

One can each intermediate field L of the extension to assign the subgroup of the Galois group, the element-wise fixed leaves L, and conversely, each subgroup H of the intermediate body, which they fixed. To Hungerford (1981) is used herein to both mappings are both referred to as Galoiskorrespondenz using " priming " notation:

For intermediate fields L and M of the extension, subgroups H and J of G the following relations hold:

The field extension is called here a Galois extension if it is normal and separable. This is exactly the case if and only if, so if fixes the Galois group except the body no further elements of F. As is true in all cases, the extension is Galois if and only if is. The same condition applies to intermediate body L: The extension is exactly then a Galois extension if and only if. The terms normal and separable be defined independently of the mappings used here in the article body extension. There Galois extension is defined the same in the event the section that the extension is algebraic. The definition used here also allows for non- algebraic extensions by Emil Artin and Hungerford (1981).

Seclusion

After Hungerford (1981 ) is called a subset X of the Galois group or an intermediate body X the extension complete if applies.

• All objects that occur as images of the correspondence described above are completed (after 6 ).
• The trivial subgroup 1, G and F are completed.
• The extension is exactly then a Galois extension if K is closed.

With the designations agreed at the beginning of the section the following applies:

• When L is completed, and finally, then M is finished and it is.
• If H is complete and is finite, then J is completed and.
• Especially applies ( for ): Every finite subgroup of the Galois group is completed.
• If F is a finite Galois extension of K, then all intermediate body and all subgroups of the Galois group are completed and the Galois group has order.

Main theorems of Galois theory

Finite Körpererweitung

If F is a finite Galoisweiterung of K, then conveys the Galoiskorrespondenz a bijection between the set of intermediate body and the amount of the subgroups of the Galois group. This correspondence forms the subset Association of intermediate body ( with order ) on the association of subgroups ( with the order > ) order-preserving from, the subset relationship is reversed. The following applies:

Infinite-dimensional algebraic extension

If F is an algebraic, not necessarily finite Galoisweiterung of K, then conveys the Galoiskorrespondenz a bijection between the set of all intermediate body and the amount of closed subgroups of the Galois group. This correspondence forms the subset Association of intermediate body ( with order ) on the association of completed sub-groups ( with the order > ) order-preserving from, the subset relationship is reversed. The following applies:

Examples

• The complex numbers are a body and contain the field of real numbers. So is a field extension. As a vector space of dimension 2 over ( a base ), applies. The Galois group contains the identity and complex conjugation. The root set of the minimal polynomial is. The identity of these two elements is back on themselves, while they are permuted by the complex conjugation. So the Galois group is restricted to the root set isomorphic to the symmetric group
• Be, the field of rational functions over K. Then for every number by the defined mapping a K- automorphism. If the field K is infinite, then there are infinitely many of these automorphisms and the Galois group is an infinite group. If the element itself has no unit root, then the heat generated by the automorphism group of G is not complete.
• The field of real numbers does not allow non-trivial automorphisms, because his arrangement is an algebraic invariant: It's for two real numbers if and only if is a square. Therefore, the field of real numbers on any of his proper subfield is Galois, the same applies to the field of real algebraic numbers.
• More generally this applies to all Euclidean body: the Galois group of a Euclidean body over a part of its body is always the trivial group.

Galois group of a cubic polynomial

The following detailed example illustrates the polynomial as intermediate body can be determined by the Galois group.

The number produced by the body over real number has the Galois group 1 because no other zeros of the minimal polynomial of ξ in ( reellen! ) Number bodies lie. This extension is therefore not Galois. Your degree is 3, since isomorphic to the factor ring (see factor ring). The same applies for the two numbers, and the body which are not of real roots of the two F, or be generated. All three bodies are isomorphic intermediate body of the splitting field F of the polynomial f

Since the body is perfect as a body of characteristic 0, the splitting is looking for a Galois extension of the Galois group G and must operate transitively on the roots of f. The only proper subgroup of the symmetric group which acts transitively on, is the normal subgroup generated by the 3- cycle of the alternating group. As we have already identified three real intermediate body and has no proper subgroups, it can not be the full Galois group itself. So this can only be the full symmetric group. In addition to the intermediate bodies, which we have already identified, nor a normal intermediate body e must be present, which is two-dimensional over ( index ). This remains fixed under cyclic permutations of the zeros, which applies only to the cyclotomic field of the third roots of unity, which is generated by the unit root. All results are shown in the diagram below.

The intermediate body can now be used, among other things, to win various representations of the splitting field:

• , This follows - without Galois theory - from its definition as a splitting field.
• : That satisfy two zeros for the generation, follows from the fact that there are no other body between the bodies, which are generated by a zero and F.
• Here (in this case single maximum ) Subnormalreihe the Galois group is simulated ( in the graph of the path right). The relative enhancements are all Galois and its Galois groups are abelian simple groups.
• F can be represented as a simple field extension: is certainly an element of F and is fixed by any nontrivial element of the Galois group. Therefore.

Of course, the zeros can be freely exchanged in all these representations.