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)
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is an ideal in . In particular, every element in can be written as a fraction, with a fixed denominator.
(2)
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Note that the multiplication of two fractional ideals is another fractional ideal.
For example, in the field , the set
(3)
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is a fractional ideal because
(4)
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Note that , where
(5)
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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.