In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, quadratic integers are a generalization of the usual
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s to
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...
s. A complex number is called a quadratic integer if it is a
root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
of some
monic polynomial
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. That is to say, a monic polynomial is one ...
(a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
whose
leading coefficient is 1) of degree two whose coefficients are integers, i.e. quadratic integers are
algebraic integer
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s of degree two. Thus quadratic integers are those complex numbers that are 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 root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
s of rational integers, such as
, and the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
, which generates the
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf ...
s. Another common example is the non-
real cubic
root of unity
In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...
, which generates the
Eisenstein integer
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
: z = a + b\omega ,
where and are integers and
: \omega = \frac ...
s.
Quadratic integers occur in the solutions of many
Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name:
*Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real n ...
s, such as
Pell's equations, and other questions related to integral
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
s. The study of rings of quadratic integers is basic for many questions of
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
.
History
Medieval
Indian mathematicians
Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians in the modern era. One of such works is Hindu numeral system which is predominantly used today and is likely ...
had already discovered a multiplication of quadratic integers of the same
discriminant , 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
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
of degree two. More explicitly, it is a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
, which solves an equation of the form , with and
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s. Each quadratic integer that is not an integer is not
rational
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
– namely, it's a real
irrational number
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
if and non-real if – and lies in a uniquely determined
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...
, 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
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
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
In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...
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
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
as a complex number (this is false if
). The norm is a
completely multiplicative function
In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime ...
, 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
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all 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
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
consisting of the
algebraic integer
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s contained in
It is the set
where
if , and otherwise. It is often denoted
, because it is the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...
of
, which is the
integral closure
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over a subring ''A'' of ''B'' if ''b'' is a root of some monic polynomial over ''A''.
If ''A'', ''B'' are fields, then the notions of "integral over" and ...
of
in
The ring