Minkowski bound for ideal classes

By [F1] the ring ${\mathcal{O}}_K$ is Dedekind, so the fractional ideals and the class group of [F2] and [F3] are available, and the class $[I]$ has a nonzero fractional representative $I$.

By the denominator condition in [F2] applied to the fractional ideal $I^{-1}$, there is $0\mathrm{\ne }u\in {\mathcal{O}}_K$ with $\mathfrak{b}:=u{I}^{-1}\subseteq {\mathcal{O}}_K$; $\mathfrak{b}$ is a nonzero integral ideal, and $[\mathfrak{b}]=[I{]}^{-1}$ in $\mathrm{Cl}({\mathcal{O}}_K)$ because $u$ contributes the principal class.

Applying [F4] to the nonzero integral ideal $\mathfrak{b}$ gives $0\mathrm{\ne }\beta \in \mathfrak{b}$ with $\mathrm{\mid }{N}_{K\mathrm{/}\mathbb{Q}}(\beta )\mathrm{\mid }\le M_{K}N\mathfrak{b}$.

By [F6] the membership $\beta \in \mathfrak{b}$ says $(\beta )\subseteq \mathfrak{b}$; multiplying this inclusion by the fractional ideal ${\mathfrak{b}}^{-1}$ and using $\mathfrak{b}{\mathfrak{b}}^{-1}={\mathcal{O}}_K$ from [F2] gives $\mathfrak{a}:=(\beta ){\mathfrak{b}}^{-1}\subseteq {\mathcal{O}}_K$, a nonzero integral ideal because $(\beta )\mathrm{\ne }0$ and ${\mathfrak{b}}^{-1}\mathrm{\ne }0$.

In the class group, $[\mathfrak{a}]=[(\beta )][\mathfrak{b}{]}^{-1}=[\mathfrak{b}{]}^{-1}=[I]$, since the principal fractional ideal $(\beta )$ represents the identity class.

From $\mathfrak{a}=(\beta ){\mathfrak{b}}^{-1}$ we get the identity of integral ideals $\mathfrak{a}\mathfrak{b}=(\beta )$; both factors are nonzero integral ideals, so [F5] gives $N\mathfrak{a}N\mathfrak{b}=N((\beta ))=\mathrm{\mid }{N}_{K\mathrm{/}\mathbb{Q}}(\beta )\mathrm{\mid }\le M_{K}N\mathfrak{b}$, and dividing by the positive integer $N\mathfrak{b}$ yields $N\mathfrak{a}\le M_K$.

Thus the integral ideal $\mathfrak{a}$ lies in the class $[I]$ and satisfies $N\mathfrak{a}\le M_K$; since the class was arbitrary, every class of $\mathrm{Cl}({\mathcal{O}}_K)$ contains such an ideal.