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 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.

### Other articles related to "ideal, principal ideal":

... 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 ...

... 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**is an unincorporated community in northern Tripp County, South Dakota, United States, named for its superb farmland ...

... 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)