Teorema de Kronecker-Weber

In the algebraic theory of numbers, the Kronecker-Weber Theorem states that every finite abelian extension of the body of rational numbers Q {\displaystyle {\mathbb {Q}}} , or in other words each body of algebraic numbers whose group of Galois on Q {\displaystyle {\mathbb {Q}}} is abelian, is a subbody of a cyclotomic body, ie a body obtained by adding a root of unity to rational numbers. The German mathematician Leopold Kronecker provided most of the test in 1853, whose gaps were filled by Weber in 1886 and Hilbert in 1896. It can be proved by direct algebraic construction, but it is also a simple consequence of the theory of bodies of classes and can be tested by collecting local data on the p-adic field of each prime p.

For an abelian extension K of Q there is in fact a minimum cyclotomic field containing it. The theorem allows one to define the conductor fde K, as the smallest integer n such that K resides in the body generated by the nth roots of the unit. For example, quadratic bodies have the absolute value of their discriminant as a conductor, a fact widely generalized in class theory.

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