Theory of Artin-Schreier
In mathematics, the theory of Artin-Schreier gives a description of the extensions galoisiennes of degree p of a characteristic body of p . It thus treats a case inaccessible to the Théorie of Kummer.
Extension of Artin-SchreierEither K a body of characteristic p , and has an element of this body. The body of decomposition of the polynomial Xp-X+a above K , is called extension of Artin-Schreier. If B is a root of this polynomial, then all the b+i for I energy of 0 with p-1 are p - distinct roots. The body of decomposition is thus separable, K - generated by B , and a cyclic extension of degree p of K , a generator of the group of Welshman being the morphism defined by .
For example, the Corps finished with two elements admits like extension of Artin-Schreier the body finished to 4 elements, generated by the polynomial X2-X+1=X2+X+1 .
Theory of Artin-SchreierThe theory of Artin-Schreier consists of reciprocal with the fact below: any cyclic extension of degree p of a body of characteristic p is an extension of Artin-Schreier. This is shown for example by using the Théorème 90 of Hilbert under its additive version.
The extensions of degree p not galoisiennes cannot be described using this theory. For example, extension obtained by adding a root p - ème of unspecified the T (i.e. a root of the polynomial in unspecified the X , Xp-T , which is inseparable) in the body of the functions to a variable above the body first with p elements.
A theory similar in characteristic p to that of the resolution by radicals, must thus authorize extensions of Artin-Schreier. To obtain extensions of order of the powers of the characteristic, it is necessary to use the theory of the vectors of Witt.
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