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. By the principal ideal theorem, any non-principal 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 non-principal 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 {\displaystyle y} be a root of y 2 + y + 6 = 0 {\displaystyle y^{2}+y+6=0} , then the ring of integers of the field Q ( y ) {\displaystyle \mathbb {Q} (y)} is Z [ y ] {\displaystyle \mathbb {Z} [y]} , which means all a + b ⋅ y {\displaystyle a+b\cdot y} with a {\displaystyle a} and b {\displaystyle b} integers form the ring of integers. An example of a nonprincipal ideal in this ring is the set of all 2 a + y ⋅ b {\displaystyle 2a+y\cdot b} where a {\displaystyle a} and b {\displaystyle 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 {\displaystyle w} satisfying w 3 − w − 1 = 0 {\displaystyle w^{3}-w-1=0} to Q ( y ) {\displaystyle \mathbb {Q} (y)} , giving Q ( y , w ) {\displaystyle \mathbb {Q} (y,w)} . An ideal number for the nonprincipal ideal 2 a + y ⋅ b {\displaystyle 2a+y\cdot b} is ι = ( − 8 − 16 y − 18 w + 12 w 2 + 10 y w + y w 2 ) / 23 {\displaystyle \iota =(-8-16y-18w+12w^{2}+10yw+yw^{2})/23} . Since this satisfies the equation
ι 6 − 2 ι 5 + 13 ι 4 − 15 ι 3 + 16 ι 2 + 28 ι + 8 = 0 {\displaystyle \iota ^{6}-2\iota ^{5}+13\iota ^{4}-15\iota ^{3}+16\iota ^{2}+28\iota +8=0} it is an algebraic integer. All elements of the ring of integers of the class field which when multiplied by ι {\displaystyle \iota } give a result in Z [ y ] {\displaystyle \mathbb {Z} [y]} are of the form a ⋅ α + y ⋅ β {\displaystyle a\cdot \alpha +y\cdot \beta } , where
α = ( − 7 + 9 y − 33 w − 24 w 2 + 3 y w − 2 y w 2 ) / 23 {\displaystyle \alpha =(-7+9y-33w-24w^{2}+3yw-2yw^{2})/23}
and
β = ( − 27 − 8 y − 9 w + 6 w 2 − 18 y w − 11 y w 2 ) / 23. {\displaystyle \beta =(-27-8y-9w+6w^{2}-18yw-11yw^{2})/23.}
The coefficients α and β are also algebraic integers, satisfying
α 6 + 7 α 5 + 8 α 4 − 15 α 3 + 26 α 2 − 8 α + 8 = 0 {\displaystyle \alpha ^{6}+7\alpha ^{5}+8\alpha ^{4}-15\alpha ^{3}+26\alpha ^{2}-8\alpha +8=0}
and
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