Class Group (original) (raw)
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Let 
be a number field, then each fractional ideal 
of 
belongs to an equivalence class ![[I]](http://mathworld.wolfram.com/images/equations/ClassGroup/Inline4.svg)
consisting of all fractional ideals 
satisfying 
for some nonzero element 
of 
. The number of equivalence classes of fractional ideals of 
is a finite number, known as the class number of 
. Multiplication of equivalence classes of fractional ideals is defined in the obvious way, i.e., by letting ![[I][J]=[IJ]](http://mathworld.wolfram.com/images/equations/ClassGroup/Inline11.svg)
. It is easy to show that with this definition, the set of equivalence classes of fractional ideals form an Abelian multiplicative group, known as the class group of 
.
See also
Class Number, Equivalence Class, Fractional Ideal
This entry contributed by David Terr
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References
Marcus, D. A. Number Fields, 3rd ed. New York: Springer-Verlag, 1996.
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Cite this as:
Terr, David. "Class Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.https://mathworld.wolfram.com/ClassGroup.html