Quadratic integer  From Wikipedia, the free encyclopedia Jump
to: navigation, search  In number theory, quadratic integers are
a generalization of the rational integers to quadratic fields.
These are algebraic integers of the degree 2. Important examples
include the Gaussian integers and the Eisenstein integers.
Though they have been studied for more than a hundred years,
many open problems remain.   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 = \sqrt{D} & \mbox{if }D
\equiv 2, 3 \pmod{4} \\ {{1 + \sqrt{D}} \over 2} & \mbox{if }D
\equiv 1 \pmod{4}  (D is a square-free integer. Note that the
case  D \equiv 0\pmod{4} 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.[1][2]  The set of all
quadratic integers is not