Artin–Schreier theory

The Artin - Schreier theory belongs in mathematics to field theory. For body positive characteristic describes abelian Galois extensions from the exponent and thus adds to the grief theory. It is named after Emil Artin and Otto Schreier.

Motivation: cyclic extensions of degree p

Be a field of characteristic. The starting point of the Artin - Schreier theory is the Artin - Schreier polynomial

For one. From Fermat's Little Theorem or abstract from the properties of Frobeniushomomorphismus follows: For being. This is why: If a zero of in an extension field of, then the other zeros. Has no zeros in, hence it is irreducible, and the extension field is Galois with Galois group generated by.

Conversely, suppose that a Galois extension of degree and a producer of the Galois group. After the normal basis exists a set, so that is a basis of a vector space. By construction, the track

Not 0 Set

Then

In order to

This is invariant under the Galois group, that is.

The so constructed element depends on the choice of starting, but in a controlled manner: If another element, then, that is with an element, and

Consequently, the residue class of modulo is uniquely determined.

Results

Be a field of characteristic.

• Be. The mapping that assigns an element of the splitting field of the polynomial induces a bijection from the set of isomorphism classes of Galois extensions of degree.

The more general version of Ernst Witt is:

• Be a separable degree of and additive group homomorphism. Then there is the following explicit bijection between the set of subgroups of and the amount of (not necessarily finite) abelian extensions of exponent (ie, for each element of the Galois group applies ): A subset of 'll identify with their archetype. Then the corresponding abelian extension of exponent. For finite subgroups. The inverse map assigns an extension to the group.

Galoiskohomologische interpretation

Be continues to be a field of characteristic, a separable degree of and. Be also the absolute Galois group of. The polynomial for each separable because its derivative is. Therefore, the homomorphism is surjective. Its core is. One thus gets a short exact sequence of -modules:

It induces in the Galoiskohomologie a long exact sequence

It was used:

• (continuous homomorphisms ) because trivial operates on
• Because over all finite Galois extensions is. With a generalization of the above argument with the normal base rate, one can show.

For the consideration of extensions of degree, the general statement is not required: Let be a Galois extension of degree. Then, and by linking with the projection one obtains a homomorphism. With the embedding gives a 1- cocycle, which is however already in the subgroup. The above- constructed element has the property for all, that is a 1- coboundary. The general gruppenkohomologische construction shows that a pre-image of being under the Verbindungshomomorphismus.

Is right reversed, you can choose an archetype, and the homomorphism. The core of and correspond to each other under the Galois correspondence.

So the resulting from the long exact sequence isomorphism is identical to the above-explained explicit construction.

For the more general statement of sub-groups one must still subgroups of extensions with the exponent identify: One subgroup corresponds to the fixed field of, an abelian extension of exponent corresponds to the subgroup of homomorphisms that factorize through the quotient.

Artin - Schreier symbol and class field theory

The Artin - Schreier symbol is an addition to power residue symbol and is used like this, the explicit description of the local Reziprozitätsabbildung and thus leads to a partial statement of the existence theorem of local class field theory. Be a local field of characteristic, ie isomorphic to a formal Laurent series body for power. The Artin - Schreier symbol arises from the cohomological pairing

By concatenation with the Reziprozitätsabbildung. If and and, then:

The Artin - Schreier symbol induces a nondegenerate bilinear form

Other features include:

• It is precisely when a norm in the extension is.
• It applies to everyone.

The Artin - Schreier symbol has the following explicit description: Be a symbol of the one-dimensional, spanned by the vector space and

And the Residuenabbildung

( The construction is independent of the isomorphism. ) For and is then:

From this formula one can prove that the Artin - Schreier symbol as alleged is not degenerate. It follows that an item that is for every Galois extension of degree in the standard group, a- th power. It follows that the intersection of all norm groups is trivial, an essential step (depending on access ) in the proof of the local existence theorem.

The local Artin - Schreier - symbols can be combined to a global pairing

(where the needle ring and the Idelgruppe ) together and use them for the proof of the global existence theorem in the function field case.

Geometric view

In the center of the geometric point of view is the Artin - Schreier - morphism

Can be regarded as a long - isogeny of the additive group ( the relative Frobeniusmorphismus ). is a ( coherent and therefore non-trivial ) étale Galois overlay with group. The existence of showing that the geometric étale fundamental group of the affine line is not trivial, in contrast to the situation in characteristic 0

A body member corresponds to a morphism, and the fiber in excess of either the trivial - Torsor or defined by the polynomial expansion of Artin Schreier.

For Artin - Schreier - Torsor associated sheaves are relevant for the Fourier - Deligne transform.

Artin - Schreier -Witt theory

The theory outlined here generalizes the Artin - Schreier theory to extensions whose exponent is a power of. It is the content of the work of Witt, in which he introduces the Witt vectors. The first part is a general statement about abelian extensions of fields of characteristic, the second part of an explicit description of a part of the local class field theory in the case of function fields.

Be back a field of characteristic, a separable and conclusion of the absolute Galois group of. Is the group of the typical Witt vectors of length and Frobeniushomomorphismus

With

Is

An exact sequence of -modules, having been used. Galois cohomology disappears because the ratio of the filtration are isomorphic with respect to and is (see above). So, and as above, one obtains a correspondence between abelian extensions, whose exponent is a divisor of, and subgroups of.

Be a local body (formal Laurent series ). At a Witt vector and a body member defines a central simple algebra Witt, the from and the commutating elements with the relations

Is generated. This is expected as a Witt vector, and is the Witt vector. Be with and, besides, the Reziprozitätsabbildung. The Artin - Schreier -Witt symbol is defined as

The Artin - Schreier -Witt symbol is a nondegenerate bilinear pairing

It is precisely when applies. The value of the symbol is equal to the invariant of the central simple algebra. Witt is also a description of the invariant as a continuing on Witt vectors of Laurent series residual.