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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. ==Example== For instance, let ''y'' be a root of ''y''2 + ''y'' + 6 = 0, then the ring of integers of the field is , 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 2''a'' + ''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 ''w''3 − ''w'' − 1 = 0 to , giving . An ideal number for the nonprincipal ideal 2''a'' + ''yb'' 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 ι give a result in are of the form ''a''α + ''b''β, where : and : The coefficients α and β are also algebraic integers, satisfying : and : respectively. Multiplying ''a''α + ''b''β by the ideal number ι gives 2''a'' + ''by'', which is the nonprincipal ideal. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「ideal number」の詳細全文を読む スポンサード リンク
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