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Fractional Ideal

المؤلف:  Atiyah, M. and MacDonald, I.

المصدر:  Ch. 9 in Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969.

الجزء والصفحة:  ...

17-10-2019

1372

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

{bx such that x in f} " src="http://mathworld.wolfram.com/images/equations/FractionalIdeal/NumberedEquation1.gif" style="height:15px; width:167px" />

(1)

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

{a/b such that a in a} " src="http://mathworld.wolfram.com/images/equations/FractionalIdeal/NumberedEquation2.gif" style="height:15px; width:139px" />

(2)

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

For example, in the field , the set

{(2a_1+a_2-5a_4+(a_2+2a_3+a_4)sqrt(-5))/(3+sqrt(-5)) such that a_i in Z} " src="http://mathworld.wolfram.com/images/equations/FractionalIdeal/NumberedEquation3.gif" style="height:94px; width:266px" />

(3)

is a fractional ideal because

(4)

Note that , where

{3b_1+b_2-5b_4+(b_2+3b_3+b_4)sqrt(-5) such that b_i in Z}=<3,1+sqrt(-5)>, " src="http://mathworld.wolfram.com/images/equations/FractionalIdeal/NumberedEquation5.gif" style="height:47px; width:258px" />

(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.

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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.