Galois theory
The Galois theory is a branch of algebra. In classical point of view, the Galois theory deals with the symmetries of the zeros of polynomials (which are the solutions or roots of the associated polynomial equation ). These symmetries may in principle by groups of permutations, that sub-groups of the symmetric group, are described. Évariste Galois discovered that these symmetries allow statements about the solvability of the equation. In modern visual field extensions are examined with the help of its Galois group.
The Galois theory has many applications to classical problems, such as " Which regular polygons can be constructed with ruler and compass? ", " Why can an angle can not be trisected? " (Again, only with a compass and ruler) and "Why there is no closed formula for the calculation of the zeros of polynomials fifth or higher degree, which does only the four basic arithmetic operations and square roots? "( the theorem of Abel - Ruffini ).
- 2.1 Fundamental Theorem of Galois Theory
Conventional approach
A " symmetry of the zeros of polynomials " is a permutation of the zeros, so that every algebraic equation over these zeros and then is still valid after you have the zeros reversed. These permutations form a group. Depending on the coefficients that are allowed in the algebraic equations, different Galois groups.
Galois himself described a method by which an individual fulfilled by the zeros of the equation can be constructed ( the so-called Galois resolvent ), so that the Galois group consists of the symmetries of an equation.
Example
The Galois group of the polynomial to be determined over the field of rational numbers. So that only rational numbers in the algebraic equations, which are filled with zeros, as coefficients allows.
The zeros of the polynomial
There are ways to permute these four zeros ( to swap ), but not all of these permutations also belong to the Galois group. This is because that all algebraic equations with only rational coefficients which contain the variables a, b, c and d, also under the permutations of the Galois group must retain their validity. For example, looking, so this equation is correct, if we interchange any of the zeros of each other. Of the permutation, the a and b can be the same, and c and d are interchanged, is formed in the equation but a false statement. Because a is mapped to a and d to c, but is not equal to 0 Therefore, this permutation is not part of the Galois group.
Another equation that satisfy the zeros is. Therefore, we can reflect on, because we have them too. But we can not reflect on, there. On the other hand, we can map, although, and as the equation as the coefficient has an irrational number, so that this equation is relevant for the definition of the Galois.
All these requirements eliminate permutations of the Galois group, so this ultimately contains only the following four permutations and isomorphic to the Klein four-group is:
Or in cycles notation:
Modern approach
The modern approach formulates the Galois theory in the language of algebraic structures: Starting from a field extension L / K we define the Galois group as the group of all Körperautomorphismen of L, which hold the elements of K separately.
Where L is a smallest extension field K, in which the given polynomial is divided into linear factors. It is called normal or Galois extension field of K. The Galois group consisting of those automorphisms of L which leave the lower body K elementwise fixed, so lets also necessary to identify each term whose value is an element of K.
In the example above, we compute the Galois group of the field extension Q (a, b, c, d) / Q.
The knowledge of solvable groups in group theory allows us to determine if a polynomial is solvable by radicals, depending on whether the Galois group is solvable or not. Each field extension L / K belongs to a factor group of the main sequence of the Galois group. If a factor group of the main sequence is cyclic of order n, the corresponding field extension is a radical extension, and the elements of L can be used as the n- th roots of an element of K to be construed.
If all factor groups of the main sequence are cyclic, the Galois group is called solvable, and all the elements of the associated body may by successive extraction of roots, product form and summing of the elements of the body ( usually Q) can be obtained.
One of the greatest triumphs of Galois theory was the proof that a polynomial exists for any n > 4 with degree n, which is not solvable by radicals. This is due to the fact that, for n> 4, the symmetric group Sn contains a simple acyclic normal divider.
Fundamental theorem of Galois theory
If L is a finite Galois extension of the field K, and G ( L / K ) be the corresponding Galois group, then L is Galois over every intermediate field Z, and there is an inclusive reversing bijection
Normal body extensions correspond under this bijection normal subgroups of. In addition:
A more general formulation is explained in the article Galois group.
Generalizations
In the case of an infinite expansion can be the automorphism group of the so-called Krull topology provided ( by W. Krull ). Is separable and normal ( ie, a Galois extension ), then there is a natural bijection between partial extensions and closed subsets of.
Is not necessarily algebraic infinite extension, so there is no such general theory more: For example, if a perfect field of characteristic, then by
A Körperautomorphismus defined, the so-called Frobeniushomomorphismus. The subgroup of generated by is generally "much " smaller than the group of automorphisms of, but it is true. If an algebraic closure of, so, however, the subgroup generated by the Frobenius is dense in, ie its degree is equal to the Galois group.
However, if a field extension with (that does not imply that L / K is algebraically and thus in particular is not a Galois field ), so still applies: and are inverse to each other, for inclusion -reversing bijections between the set of compact subsets of and the amount of the intermediate bodies in which L is Galois over M.
There is also a generalization of the Galois theory for ring extensions instead of field extensions.
The inverse problem of Galois theory
It is easy to construct field extensions with any given finite group as Galois group if one does not set the base body. Therefore all finite groups occur as Galois group.
You simply select a body and a finite group. By the theorem of Cayley is isomorphic to a subgroup of the symmetric group on the elements of. If one chooses variables for each element of and adjoint them, we obtain. In the body include the symmetric rational functions in the is. Then, and the fixed field below has Galois group after the main theorem of Galois theory.
However, there is a generally unsolved problem of how and whether one. , Such a construction for a solid body, such as, can perform