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Fractional Ideal


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

 a=bf={bx such that x in f}
(1)

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

 f={a/b such that a in a}
(2)

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

For example, in the field Q(sqrt(-5)), the set

 f={(2a_1+a_2-5a_4+(a_2+2a_3+a_4)sqrt(-5))/(3+sqrt(-5)) 
  such that a_i in Z}
(3)

is a fractional ideal because

 (3+sqrt(-5))f=<2,1+sqrt(-5)>.
(4)

Note that fp=<1>=R, where

 p={3b_1+b_2-5b_4+(b_2+3b_3+b_4)sqrt(-5) 
  such that b_i in Z}=<3,1+sqrt(-5)>,
(5)

and so f is an inverse to p.

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


See also

Class Group, Grothendieck Group, Ideal, Number Field

This entry contributed by Todd Rowland

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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.

Referenced on Wolfram|Alpha

Fractional Ideal

Cite this as:

Rowland, Todd. "Fractional Ideal." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FractionalIdeal.html

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