Principal ideal theorem

Principal ideal theorem

:"This article is about the Hauptidealsatz of class field theory. You may be seeking Krull's principal ideal theorem, also known as Krull's Hauptidealsatz, in commutative algebra"

In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, is the statement that 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

:"IO""L"

is a principal ideal α"O""L", for "O""L" the ring of integers of "L" and some element α in it. In other terms, extending ideals gives a mapping on the class group of "K", to the class group of "L", which sends all ideal classes to the class of a principal ideal. The phenomenon has also been called "principalization", or sometimes "capitulation". It was conjectured by David Hilbert, and was the last remaining aspect of his programme on class fields to be completed, around 1930.

The question was reduced to a piece of finite group theory by Emil Artin. That involved the transfer. The required result was proved by Philipp Furtwängler.

References

*Ph. Furtwängler, "Beweis des Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper", Abh. Math. Sem. Hamburg 7 (1930).


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