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 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 √2, and the complex number i = √–1, which generates the Gaussian integers. Another common example is the non-real cubic root of unity −1 + √–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. The characterization of the quadratic integers was first given by Richard Dedekind in 1871. 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={frac {-Bpm {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 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. D may be positive or negative. The quadratic integers (including the ordinary integers), which 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}}).} Here and in the following, D is supposed to be a square-free integer. This does not restrict the generality, as the equality √a2D = a√D (for any positive integer a) implies Q ( D ) = Q ( a 2 D ) . {displaystyle mathbb {Q} ({sqrt {D}})=mathbb {Q} ({sqrt {a^{2}D}}).}