Quadratic integer

In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form

with $b$ and $c$ (usual) integers. When algebraic integers are considered, the usual integers are often called rational integers.

Common examples of quadratic integers are the square roots of rational integers, such as $\sqrt 2$ , and the complex number $i = \sqrt -1$ , which generates the Gaussian integers. Another common example is the non-real cubic root of unity $-1 + \sqrt -3 / 2$ , which generates the Eisenstein integers.

Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory.

Definition[edit]

A quadratic integer is an algebraic integer of degree two. More explicitly, it is a complex number $x=(-b\pm {\sqrt {b^{2}-4c}})/2$ , which solves an equation of the form $x 2 + bx + c = 0$ , with $b$ and $c$ integers. Each quadratic integer that is not an integer is not rational – namely, it's a real irrational number if $b 2 - 4 c > 0$ and non-real if $b 2 - 4 c < 0$ – and lies in a uniquely determined quadratic field $\mathbb {Q} ({\sqrt {D}}\,)$ , the extension of $\mathbb {Q}$ generated by the square root of the unique square-free integer $D$ that satisfies $b 2 - 4 c = De 2$ for some integer $e$ . If $D$ is positive, the quadratic integer is real. If $D < 0$ , it is imaginary (that is, complex and non-real).

The quadratic integers (including the ordinary integers) that belong to a quadratic field $\mathbb {Q} ({\sqrt {D}}\,)$ form an integral domain called the ring of integers of $\mathbb {Q} ({\sqrt {D}}\,).$

Although the quadratic integers belonging to a given quadratic field form a ring, the set of all quadratic integers is not a ring because it is not closed under addition or multiplication. For example, $1+{\sqrt {2}}$ and ${\sqrt {3}}$ are quadratic integers, but $1+{\sqrt {2}}+{\sqrt {3}}$ and $(1+{\sqrt {2}})\cdot {\sqrt {3}}$ are not, as their minimal polynomials have degree four.

A classic example is $\mathbf {Z} [{\sqrt {-1}}\,]$ , the , which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law.^[5]

Gaussian integers

The elements in ${\mathcal {O}}_{\mathbf {Q} ({\sqrt {-3}}\,)}=\mathbf {Z} \left[{{1+{\sqrt {-3}}} \over 2}\right]$ are called .