Fractional Ideal

A fractional ideal is a generalization of an ideal in a ring . Instead, a fractional ideal is contained in the number field , but has the property that there is an element  such that

(1)

is an ideal in . In particular, every element in  can be written as a fraction, with a fixed denominator.

(2)

Note that the multiplication of two fractional ideals is another fractional ideal.

For example, in the field , the set

(3)

is a fractional ideal because

(4)

Note that , where

(5)

and so  is an inverse to .

Given any fractional ideal  there is always a fractional ideal  such that . Consequently, the fractional ideals form an Abelian group by multiplication. The principal ideals generate a subgroup , and the quotient group is called the ideal class group.

REFERENCES:

Atiyah, M. and MacDonald, I. Ch. 9 in Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.

Cohn, H. Introduction to the Construction of Class Fields. New York: Cambridge University Press, p. 32, 1985.

Fröhlich, A. and Taylor, M. Ch. 2 in Algebraic Number Theory. New York: Cambridge University Press, 1991.