# Principal Ideal Theorem

class="left_top_holder">

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":

Cardinal Point (optics) - Modeling Optical Systems As Mathematical Transformations - Ideal, Rotationally Symmetric, Optical Imaging System
... 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⟩ ...
Ideal, South Dakota
... Ideal is an unincorporated community in northern Tripp County, South Dakota, United States, named for its superb farmland ...
Leverage
... An ideal mechanism transmits power without adding to or subtracting from it ... This means the ideal mechanism does not include a power source, and is frictionless and constructed from rigid bodies that do not deflect or wear ... of real systems is obtained from this ideal by using efficiency factors that take into account friction, deformation and wear ...

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

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)

The principal thing children are taught by hearing these lullabies is respect. They are taught to respect certain things in life and certain people. By giving respect, they hope to gain self-respect and through self-respect, they gain the respect of others. Self-respect is one of the qualities my people stress and try to nurture, and one of the controls an Indian has as he grows up. Once you lose your self-respect, you just go down.
Henry Old Coyote (20th century)

But I must needs take my petulance, contrasting it with my accustomed morning hopefulness, as a sign of the ageing of appetite, of a decay in the very capacity of enjoyment. We need some imaginative stimulus, some not impossible ideal which may shape vague hope, and transform it into effective desire, to carry us year after year, without disgust, through the routine- work which is so large a part of life.
Walter Pater (1839–1894)