In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree two whose coefficients are integers, i.e. quadratic integers are algebraic integers 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 square roots of rational integers, such as 2 {\textstyle {\sqrt {2}}} , and the complex number i = − 1 {\textstyle i={\sqrt {-1}}} , which generates the Gaussian integers. Another common example is the non-real cubic root of unity − 1 + − 3 2 {\textstyle {\frac {-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.
History
Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same discriminant D, which allowed them to solve some cases of Pell's equation. The characterization given in § Explicit representation of the quadratic integers was first given by Richard Dedekind in 1871.
Definition A quadratic integer is an algebraic integer of degree two. More explicitly, it is a complex number x = ( − b ± b 2 − 4 c ) / 2 {\displaystyle x=(-b\pm {\sqrt {b^{2}-4c}})/2} , which solves an equation of the form x2 + bx + c = 0, with b and c integers. Each quadratic integer that is not an integer is not rational—namely, it is a real irrational number if b2 − 4c > 0 and non-real if b2 − 4c < 0—and lies in a uniquely determined quadratic field Q ( D ) {\displaystyle \mathbb {Q} ({\sqrt {D}}\,)} , the extension of Q {\displaystyle \mathbb {Q} } generated by the square root of the unique square-free integer D that satisfies b2 − 4c = De2 for some integer e. If D is positive, the quadratic integer is real. If D < 0, it is imaginary (that is, complex and non-real). The quadratic integers (including the ordinary integers) that belong to a quadratic field Q ( D ) {\displaystyle \mathbb {Q} ({\sqrt {D}}\,)} form an integral domain called the ring of integers of Q ( D ) . {\displaystyle \mathbb {Q} ({\sqrt {D}}\,).}
Although the quadratic integers belonging to a given quadratic field form a ring, the set of all quadratic integers is not a ring because it is not closed under addition or multiplication. For example, 1 + 2 {\displaystyle 1+{\sqrt {2}}} and 3 {\displaystyle {\sqrt {3}}} are quadratic integers, but 1 + 2 + 3 {\displaystyle 1+{\sqrt {2}}+{\sqrt {3}}} and ( 1 + 2 ) ⋅ 3 {\displaystyle (1+{\sqrt {2}})\cdot {\sqrt {3}}} are not, as their minimal polynomials have degree four.
Explicit representation Here and in the following, the quadratic integers that are considered belong to a quadratic field Q ( D ) , {\displaystyle \mathbb {Q} ({\sqrt {D}}\,),} where D is a square-free integer. This does not restrict the generality, as the equality a 2 D = a D {\textstyle {\sqrt {a^{2}D}}=a{\sqrt {D}}} (for any positive integer a) implies Q ( D ) = Q ( a 2 D ) . {\textstyle \mathbb {Q} ({\sqrt {D}}\,)=\mathbb {Q} ({\sqrt {a^{2}D}}\,).}
An element x of Q ( D ) {\textstyle \mathbb {Q} ({\sqrt {D}}\,)} is a quadratic integer if and only if there are two integers a and b such that either
x = a + b D , {\displaystyle x=a+b{\sqrt {D}},}
or, if D − 1 is a multiple of 4
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