Principal Ideal Theorem


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

is a principal ideal αOL, for OL 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.

Other articles related to "ideal, principal ideal":

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... An ideal, rotationally symmetric, optical imaging system must meet three criteria All rays "originating" from any object point converge to a single image point (Imaging is stigmatic) ... optical systems imaging is stigmatic for one or perhaps a few object points, but to be an ideal system imaging must be stigmatic for every object point ...
Principal Ideal
... In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R ... More specifically a left principal ideal of R is a subset of R of the form Ra = {ra r in R} a right principal ideal is a subset of the form aR = {ar r in R} a two-sided principal ideal is a subset of the form RaR = {r1 ... In that case, it is common to write the ideal generated by a as ⟨a⟩ ...
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