Abelian extension

Introduction

In general Algebra, more precisely in Theory of Welshman, a abelian extension is a Extension of body whose associated Welshman Groupe is abelian. When the group of Welshman is a cyclic Groupe, we have a cyclic extension .

Any finished extension of a Corps finished is a cyclic extension. The study of the Théorie of the bodies of classes in the case of describes in a detailed way all the abelian extensions the Corps of numbers, and Corps of algebraic curved functions of on finished bodies, like in the case of the local bodies (local theory of the body of classes).

Abelian Cyclotomy and Extensions to the top of the rational numbers

The cyclotomic extensions , obtained by the addition of roots of the unit give examples of abelian extensions to the top of any body. These extensions can be commonplace if the body which one share is Corps algebraically closed or if one associates roots 'p' my in characteristic “p”.

If the body of basic body is the body Q rational numbers, one obtains by this constructions known as the cyclotomic bodies. The body Q (I) of the numbers of Gauss a+bi , for has and B rational, is the noncommonplace example simplest. The cyclotomic bodies are the essential examples of abelian extension of Q , with the direction where any abelian extension is plunged in such a body.

Thus a maximum abelian extension Q ^ab of Q is obtained by associating all the roots of the unit with Q . It is a particular case of the theory of the local body of classes to the case of the body Q . In this case the group of Welshman of the extension of Q ^ab to the top of Q is identified with the group $\backslash \; hat\; \{\backslash \; mathbf\; \{Z\}\}\; ^\; \backslash \; star$ of the invertible elements of $\backslash \; hat\; \{\backslash \; mathbf\; \{Z\}\}$. It acts on the primitive roots NR my of the unit via its quotient $(\backslash \; mathbf\; \{Z\}\; /N\; \backslash \; mathbf\; \{Z\})\; ^\; \backslash \; star$, where k+NZ acts by sending the root $\backslash \; zeta$ on $\backslash \; zeta^k$.

Complex multiplication and Jügendtraüm

In the case of the body Q it is remarkable that a maximum abelian extension is generated by special values of the exponential function. Indeed, among the complex numbers, the roots of the unit are distinguished as those which are written $\backslash \; exp\; (2\; \backslash \; pi\; i*x)$ for the rational values of the parameter X . These parameter are the only algebraic numbers to which transcendent the $\backslash \; exp\; (2\; \backslash \; pi\; i*x)$ takes algebraic values (a root NR satisfies me the equation X^N=1 ), from where the name of special value.

In the case of bodies of quadratic numbers complex, it are necessary to add with the roots of the unit the algebraic numbers obtained starting from the coordinates of the points of torsion of an elliptic curve. It is the theory of the Multiplication complexes. These considerations fit in the Jügendtraüm of Kronecker.

Category: Algebraic theory of the numbers

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