= Unbounded element orders in imaginary quadratic ideal class groups
For every positive <integer> $n$, some <imaginary quadratic field> has an <ideal class> of <order of a group element> greater than $n$. Specifically, take a <prime number> $p\equiv1\pmod{20}$ with $p>5^n$. In $\mathbb Z[\sqrt{-p}]$, the <Dedekind factorization theorem> gives $(5)=P\overline P$, with distinct <prime ideals> $P=(5,\sqrt{-p}-2)$ and $\overline P=(5,\sqrt{-p}+2)$ of <ideal norm> five. If $P^k=(a+b\sqrt{-p})$ for $1\leq k\leq n$, the <field norm> equation $5^k=a^2+pb^2$ forces $b=0$. Then $k$ is even and $(a)=(5)^{k/2}=P^{k/2}\overline P^{k/2}$, contradicting <unique factorization of ideals in a number field>. Thus $[P]$ has <order of a group element> greater than $n$. The existence of suitable $p$ follows from <Dirichlet's theorem on arithmetic progressions>.
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