Ideal number

In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This means that there is an element of the ring of integers of the Hilbert class field, which is an ideal number, such that the original nonprincipal ideal is equal to the collection of all multiples of this ideal number by elements of this ring of integers that lie in the original field's ring of integers.

For instance, let y be a root of y² + y + 6 = 0, then the ring of integers of the field ${\displaystyle {\mathbb {Q}}(y)}$ is ${\displaystyle {\mathbb {Z}}[y]}$, which means all a + by with a and b integers form the ring of integers. An example of a nonprincipal ideal in this ring is the set of all 2a + yb where a and b are integers; the cube of this ideal is principal, and in fact the class group is cyclic of order three. The corresponding class field is obtained by adjoining an element w satisfying w³ − w − 1 = 0 to ${\displaystyle {\mathbb {Q}}(y)}$, giving ${\displaystyle {\mathbb {Q}}(y,w)}$. An ideal number for the nonprincipal ideal 2a + yb is ${\displaystyle \iota =(-8-16y-18w+12w^{2}+10yw+yw^{2})/23}$. Since this satisfies the equation ${\displaystyle \iota ^{6}-2\iota ^{5}+13\iota ^{4}-15\iota ^{3}+16\iota ^{2}+28\iota +8=0}$ it is an algebraic integer.

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