Minkowski bound for Gaussian integers

Given: The Axiom of Choice and the imaginary quadratic field $K=\mathbb{Q}(i)$ with ${\mathcal{O}}_K=\mathbb{Z}[i]$ and discriminant $d_K=-4$.

For the squarefree integer $d=-1$, which is $3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, the quadratic-field formulas give ${\mathcal{O}}_K=\mathbb{Z}[i]$ and $d_K=4\cdot (-1)=-4$ (Integers in a quadratic field, Discriminant of a quadratic field).

Signature: $r_1$ is the number of field embeddings $K\to \mathbb{R}$ fixing $\mathbb{Q}$ and $r_2$ is the number of complex-conjugate pairs among the nonreal field embeddings $K\to \mathbb{C}$ fixing $\mathbb{Q}$, with $r_1+2r_2=[K:\mathbb{Q}]$ (Archimedean embeddings and signature).

Minkowski bound: every class of $\mathrm{Cl}({\mathcal{O}}_K)$ contains an integral ideal $\mathfrak{b}$ with $N\mathfrak{b}\le M_K$ (Minkowski bound for ideal classes, The ideal class group).

For a nonzero integral ideal $\mathfrak{a}$ the norm $N\mathfrak{a}=\mathrm{\mid }{\mathcal{O}}_K\mathrm{/}\mathfrak{a}\mathrm{\mid }$ is a finite positive integer, so $N\mathfrak{a}\ge 1$, and $N\mathfrak{a}=1$ forces $\mathfrak{a}={\mathcal{O}}_K$ (The absolute norm of an integral ideal, A nonzero number-field ideal has finite quotient).

Gregory-Leibniz: $\pi \mathrm{/}4=1-1\mathrm{/}3+R_1$ with $R_1={\int }_0^1{x}^4\mathrm{/}(1+x^2)dx>0$, hence $\pi >8\mathrm{/}3>2$ (The Gregory-Leibniz series: pi over four equals 1-1/3+1/5-1/7+...).

Under the Axiom of Choice, the ring of integers of a number field is a Dedekind domain (Rings of integers are Dedekind domains).

Dedekind PID criterion: a Dedekind domain $R$ is a principal ideal domain if and only if its ideal class group is trivial (A Dedekind domain is a PID exactly when its class group is trivial).