Ring of integers of an imaginary quadratic field

Statement

For the imaginary quadratic field $K=\mathbb Q(\sqrt{-d})$ , where $d>0$ is square-free, the ring of integers is

\mathcal O_K=\begin{cases} \mathbb Z[(1+\sqrt{-d})/2],&d\equiv3\pmod4,\\ \mathbb Z[\sqrt{-d}],&d\equiv1,2\pmod4. \end{cases}

Square-freeness excludes $d\equiv0\pmod4$ .

Why the half-integer occurs

An element of a quadratic field is integral exactly when its trace and norm are integers. Write it as $(a+b\sqrt{-d})/2$ . These conditions force $a,b\in\mathbb Z$ and $a^2+db^2\equiv0\pmod4$ . Either both are even, or both are odd and $d\equiv3\pmod4$ . This gives precisely the displayed rings.

Examples

For $d=1$ this is the Gaussian integer ring. For $d=3$ it is the Eisenstein integer ring, which can equally be written $\mathbb Z[(-1+\sqrt{-3})/2]$ .

References

J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, Introduction and §2, Remark 2.12.