Quadratic integer

Definition

A quadratic integer is a complex number $x$ that satisfies an equation $x^2 + bx + c = 0$ with integers $b$ and $c$. Equivalently, $x = \dfrac{-b \pm \sqrt{b^2 - 4c}}{2}$ for some integers $b$ and $c$.

Remarks

Each integer $n$ is a quadratic integer, since it is a root of $x^2 - 2nx + n^2 = 0$. A quadratic integer that is not an integer is an algebraic number of degree $2$. It lies in a quadratic field $\mathbb{Q}(\sqrt{D})$ with $D$ a squarefree integer. The Gaussian integers are the quadratic integers in $\mathbb{Q}(i)$. Another example is the golden ratio $\frac{1 + \sqrt{5}}{2}$, a root of $x^2 - x - 1 = 0$. For $D \equiv 2, 3 \pmod 4$, the quadratic integers in $\mathbb{Q}(\sqrt{D})$ form the ring $\mathbb{Z}[\sqrt{D}] = \{ a + b\sqrt{D} : a, b \in \mathbb{Z} \}$.

Explanations

L1 human

A quadratic integer is a root of a quadratic equation with integer coefficients and leading coefficient $1$. Numbers such as $\sqrt{2}$, $i$ and $1 + \sqrt{3}$ are examples. They extend the integers in the same way that $\sqrt{2}$ extends the rationals.