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.

Famous quotes containing the words principal, ideal and/or theorem:

    As a Tax-Paying Citizen of the United States I am entitled to a voice in Governmental affairs.... Having paid this unlawful Tax under written Protest for forty years, I am entitled to receive from the Treasury of “Uncle Sam” the full amount of both Principal and Interest.
    Susan Pecker Fowler (1823–1911)

    The air was so elastic and crystalline that it had the same effect on the landscape that a glass has on a picture, to give it an ideal remoteness and perfection.
    Henry David Thoreau (1817–1862)

    To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.
    Albert Camus (1913–1960)