Let 
be a number
field, then each fractional ideal 
of 
belongs to an equivalence class ![[I]](/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]](/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 
.
Class Group
See also
Class Number, Equivalence Class, Fractional IdealThis entry contributed by David Terr
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References
Marcus, D. A. Number Fields, 3rd ed. New York: Springer-Verlag, 1996.Referenced on Wolfram|Alpha
Class GroupCite this as:
Terr, David. "Class Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ClassGroup.html