Quadratic integer

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, quadratic integers are a generalization of the usual <a href="/facts/Integer/PUwwrawx">integers</a> to <a href="/facts/Quadratic_field/dUnSDN4e">quadratic fields</a>. A complex number is called a quadratic integer if it is a <a href="/facts/Root_of_a_polynomial/mlK3dhoT">root</a> of some <a href="/facts/Monic_polynomial/4JzuIJ5R">monic polynomial</a> (a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> whose <a href="/facts/Leading_coefficient/5Hwq2Wuq">leading coefficient</a> is 1) of degree two whose coefficients are integers, i.e. quadratic integers are <a href="/facts/Algebraic_integer/Qb0fD3R3">algebraic integers</a> 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 <a href="/facts/Square_root/AfrzfBdQ">square roots</a> of rational integers, such as 
 
 
 
 
 
 2
 
 
 
 
 {\textstyle {\sqrt {2}}}
 
, and the <a href="/facts/Complex_number/2w7mqI7e">complex number</a> 
 
 
 
 i
 =
 
 
 −
 1
 
 
 
 
 {\textstyle i={\sqrt {-1}}}
 
, which generates the <a href="/facts/Gaussian_integer/zj82AhtL">Gaussian integers</a>. Another common example is the non-<a href="/facts/Real_number/R02gw5Pb">real</a> cubic <a href="/facts/Root_of_unity/EqH0ho1Y">root of unity</a> 
 
 
 
 
 
 
 −
 1
 +
 
 
 −
 3
 
 
 
 2
 
 
 
 
 {\textstyle {\frac {-1+{\sqrt {-3}}}{2}}}
 
, which generates the <a href="/facts/Eisenstein_integer/UGQmFI8c">Eisenstein integers</a>.
Quadratic integers occur in the solutions of many <a href="/facts/Diophantine_equation/mYaHp7oX">Diophantine equations</a>, such as <a href="/facts/Pell%2527s_equation/n0ul61eQ">Pell's equations</a>, and other questions related to integral <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a>. The study of rings of quadratic integers is basic for many questions of <a href="/facts/Algebraic_number_theory/LPzDT5CA">algebraic number theory</a>.

Quadratic integer open-in-new

Quadratic integer