What is "quadratic integer"

n. (context mathematics English) A number (real or complex) which is a solution to an equation of the form $x^2\; +\; Bx\; +\; C\; =\; 0$ for some integers ''B'' and ''C''. For example 42, or 2+3''i'', or $1+sqrt\{-3\}\; over\; 2$.

In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are the solutions of equations of the form

= 0

with and integers. They are thus algebraic integers of the degree two. When algebraic integers are considered, usual integers are often called rational integers.

Common examples of quadratic integers are the square roots of integers, such as , and the complex number , which generates the Gaussian integers. Another common example is the non-real cubic root of unity , which generates the Eisenstein integers.

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