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
1161
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
![a=bf=<span style=]() {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.
![f=<span style=]() {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
![f=<span style=]() {(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
![p=<span style=]() {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.
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.
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