The theorem of Kronecker-Weber establishes in Algebraic theory of the numbers the following result: each abelian Extension finished body of the rational numbers $\backslash \; mathbb\; \{Q\}$, or in other words each Bodies of numbers algebraic with Welshman a Group abelian absolute, is a subfield of a cyclotomic Extension, i.e. a body obtained by associating a Racine of the unit to the rational numbers.

This theorem was stated by Kronecker in 1853. Its proposal for a proof was incomplete. Weber in 1886 proposed a new proof, which still presented a gap. Hilbert showed it in 1896 by using methods different from those of its predecessors. The theorem is usually shown today like a consequence of the Théorie of the bodies of classes. However, it can also be deduced from the similar assertion on the p-adic Corps of numbers: if p is a prime number, and K/Q_p is a finished abelian extension, then K is included in a cyclotomic extension of Q_p .

The demonstration of the local case of the theorem requires to know well the properties of ramification of the local cyclotomic extensions, then to be reduced to the cyclic cases of extensions of order a power of a prime number Q , and to discuss according to whether q=p or not, the case p=2 in front of being still treated separately.

For a given abelian extension K of Q , there exists in fact a cyclotomic body minimal which contains it. The theorem makes it possible to define the conducting F of the K , like smallest Integer N such as K is included in the body generated by the roots N - ièmes of the unit. For example, the quadratic bodies have like driver the absolute value of their Discriminant S, a fact generalized in theory of the bodies of classes.

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