Quadratic integer

In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree two whose coefficients are integers, i.e. quadratic integers are algebraic integers of degree two. Thus quadratic integers are those complex numbers that are solutions of equations of the form

x² + bx + c = 0

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 ${\textstyle {\sqrt {2}}}$, and the complex number ${\textstyle i={\sqrt {-1}}}$, which generates the Gaussian integers. Another common example is the non-real cubic root of unity ${\textstyle {\frac {-1+{\sqrt {-3}}}{2}}}$, 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.