Quadratic Integer - Definition

Definition

Quadratic integers are solutions of equations of the form:

x2 + Bx + C = 0

for integers B and C. Such solutions have the form a + ω b , where a, b are integers, and where ω is defined by:

\omega = \begin{cases} \sqrt{D} & \mbox{if }D \equiv 2, 3 \pmod{4} \\ {{1 + \sqrt{D}} \over 2} & \mbox{if }D \equiv 1 \pmod{4} \end{cases}

(D is a square-free integer. Note that the case is impossible, since it would imply that D is divisible by 4, a perfect square, which contradicts the fact that D is square-free.).

This characterization was first given by Richard Dedekind in 1871.

The set of all quadratic integers is not closed even under addition. But for any fixed D the set of corresponding quadratic integers forms a ring, and it is these quadratic integer rings which are usually studied. Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same D, which allows one to solve some cases of Pell's equation. The study of quadratic integers admits an algebraic version: the study of quadratic forms with integer coefficients.

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