= Ideal class group of Q of square root of minus one hundred sixty-three
{c}
{title2=$\operatorname{Cl}(\mathbb Q(\sqrt{-163}))$}
The imaginary quadratic field $K=\mathbb Q(\sqrt{-163})$ has discriminant $-163$, and its <Minkowski bound for ideal classes> is
$$
\frac2\pi\sqrt{163}<9.
$$
The polynomial $X^2-X+41$ defining its <ring of integers of a quadratic field> has no root modulo $2,3,5,$ or $7$, so these rational primes are inert. Factoring an integral ideal of norm at most eight into <prime ideals> then shows that its only possible nontrivial factor is the principal ideal $(2)$, of norm four. Every ideal class is therefore principal, and
$$
\operatorname{Cl}(K)=1.
$$
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