= Solution
The <ideal class group> of a <number field> is the <group> of nonzero <fractional ideals> of its <ring of integers> under multiplication, modulo the subgroup of principal <fractional ideals>. For degree $n$ and $r_2$ pairs of complex embeddings, the <Minkowski bound for ideal classes> states that each class contains an <integral ideal> $I$ with
$$
N(I)\leq\left(\frac4\pi\right)^{r_2}\frac{n!}{n^n}\sqrt{|D_K|}.
$$
There are only finitely many <integral ideals> of bounded <ideal norm>: as sublattices of the free abelian <group> of <integers>, they have bounded index, and finitely many subgroups have any fixed index. Thus the <ideal class group> is finite.
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