Use this tag for questions related to the subset of quadratic integers contained in a quadratic field.
A quadratic integer ring, often denoted $\mathcal O_{\Bbb Q(\sqrt D)}$, is the subset of quadratic integers contained in the quadratic field $\Bbb Q(\sqrt D)$ where $D$ is a square-free integer not equal to $0$ or $1$.
Numbers of the form $a + b\sqrt D$ with rational integers $a$ and $b$ are integers of $\Bbb Q(\sqrt D)$. Those are the only integers of $\Bbb Q(\sqrt D)$ if $D \equiv 2$ or $3 \pmod 4$. If $D \equiv 1 \pmod 4$, the numbers $(c + d\sqrt D)/2$ with odd rational integers $c$ and $d$ are also integers of $\Bbb Q(\sqrt D)$, and there are no further integers.
The integers of $\Bbb Q(\sqrt{-1}) = \Bbb Q(i)$ are often called Gaussian integers.
