Normalizes (arithmetic)

In Arithmetic, the relative standard is an application of a on-body L towards a subfield K of an extension. This application intervenes in a crucial way in the Théorie of the bodies of classes: the abelian under-extensions of a given extension are primarily in correspondence with groups of standards, i.e. the image in the subfield by the standard of certain groups of on-body.

Precisely, the standard of an element of L is defined like the Déterminant K - endomorphism of multiplication by this element. It can thus also be seen like the constant term of the Polynôme characteristic of this endomorphism. In the case of an extension galoisienne, one from of deduced that the standard is the product of all combined element.

This concept extends in a concept of standard of a Idéal , definite for the ideals first like the cardinal of the residual body, then by multiplicativity, for the made up ideals. The standard of a principal ideal is then equal to the relative standard on $\ mathbb {Q}$ of a generator of this ideal. The demonstration of the finitude of the Groupe of the classes uses properties of increase of the standard of the ideals in a given class.

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