In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
with and (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 , and the complex number , which generates the Gaussian integers. Another common example is the non-real cubic root of unity , 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 , which allowed them to solve some cases of Pell's equation.
The characterization given in 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 , which solves an equation of the form , with and integers. Each quadratic integer that is not an integer is not rational – namely, it's a real irrational number if and non-real if – and lies in a uniquely determined quadratic field , the extension of generated by the square root of the unique square-free integer that satisfies for some integer . If is positive, the quadratic integer is real. If , it is imaginary (that is, complex and non-real).
The quadratic integers (including the ordinary integers) that belong to a quadratic field form an integral domain called the ring of integers of
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, and are quadratic integers, but and 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 where is a square-free integer. This does not restrict the generality, as the equality (for any positive integer ) implies
An element of is a quadratic integer if and only if there are two integers and such that either
or, if is a multiple of
with and both odd
In other words, every quadratic integer may be written , where and are integers, and where is defined by
(as has been supposed square-free the case is impossible, since it would imply that is divisible by the square 4).
Norm and conjugation
A quadratic integer in may be written
, where and are either both integers, or, only if , both halves of odd integers. The norm of such a quadratic integer is .
The norm of a quadratic integer is always an integer. If , the norm of a quadratic integer is the square of its absolute value as a complex number (this is false if ). The norm is a completely multiplicative function, which means that the norm of a product of quadratic integers is always the product of their norms.
Every quadratic integer has a conjugate
A quadratic integer has the same norm as its conjugate, and this norm is the product of the quadratic integer and its conjugate. The conjugate of a sum or a product of quadratic integers is the sum or the product (respectively) of the conjugates. This means that the conjugation is an automorphism of the ring of the integers of – see , below.
Quadratic integer rings
Every square-free integer (different from 0 and 1) defines a quadratic integer ring, which is the integral domain consisting of the algebraic integers contained in It is the set where if , and otherwise. It is often denoted , because it is the ring of integers of , which is the integral closure of in The ring consists of all roots of all equations whose discriminant is the product of by the square of an integer. In particular belongs to , being a root of the equation , which has as its discriminant.
The square root of any integer is a quadratic integer, as every integer can be written , where is a square-free integer, and its square root is a root of .