Ideal class group

Definition
Let $$K$$ be a number field (viz an algebraic extension of $$\mathbb{Q}$$ of finite degree). Let $$O$$ be the ring of integers in $$K$$ (viz the elements of $$K$$ that satisfy monic polynomials with integer coefficients). Then, the ideal class group can be defined in the following steps:


 * We can consider the set of all fractional ideals on $$O$$. A fractional ideal is a subset $$J$$ of $$K$$ such that there exists $$x \in O$$ for which $$xJ$$ is an ordinary ideal in $$O$$.
 * We can define a multiplication on fractional ideals $$I$$ and $$J$$. The product of two fractional ideals is the ideal generated by all products of elements from the two ideals. Under this multiplication, the set of all fractional ideals gets the structure of a monoid.
 * The subset of this comprising principal fractional ideals, is a submonoid
 * The quotient of the monoid of all fractional ideals by the submonoid of principal fractional ideals, turns out to be a group, and this group is the ideal class group.

Realization
Every finite Abelian group occurs as the ideal class group for some suitable choice of number field.