Principal ideal theorem

In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called principalization, or sometimes capitulation.

Formal statement

For any algebraic number field K and any ideal I of the ring of integers of K, if L is the Hilbert class field of K, then

$I{O}_L$

is a principal ideal αO_L, for O_L the ring of integers of L and some element α in it.

History

The principal ideal theorem was conjectured by Template:Harvs, and was the last remaining aspect of his program on class fields to be completed, in 1929.

Template:Harvs reduced the principal ideal theorem to a question about finite abelian groups: he showed that it would follow if the transfer from a finite group to its derived subgroup is trivial. This result was proved by Philipp Furtwängler (1929).

References

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