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Concept
Ideal number
Summary
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 be a root of , then the ring of integers of the field is , which means all with and integers form the ring of integers. An example of a nonprincipal ideal in this ring is the set of all where and 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 satisfying to , giving . An ideal number for the nonprincipal ideal is . Since this satisfies the equation
it is an algebraic integer.
All elements of the ring of integers of the class field which when multiplied by give a result in are of the form , where
and
The coefficients α and β are also algebraic integers, satisfying
and
respectively. Multiplying by the ideal number gives , which is the nonprincipal ideal.
Kummer first published the failure of unique factorization in cyclotomic fields in 1844 in an obscure journal; it was reprinted in 1847 in Liouville's journal. In subsequent papers in 1846 and 1847 he published his main theorem, the unique factorization into (actual and ideal) primes.
Official source
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