Quadratic integer

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

with $B$ and $C$ integers. When algebraic integers are considered, the usual integers are often called rational integers.

Common examples of quadratic integers are the square roots of 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.

Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same $D$ , which allowed them to solve some cases of Pell's equation.

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