In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, quadratic integers are a generalization of the usual
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (â1, â2, â3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s to
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
s. Quadratic integers are
algebraic integer
In algebraic number theory, an algebraic integer is a complex number which 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, 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
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 form ...
, 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 /ma ...
s. Another common example is the non-real cubic
root of unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
, which generates the
Eisenstein integers.
Quadratic integers occur in the solutions of many
Diophantine equations, such as
Pell's equation
Pell's equation, also called the PellâFermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...
s, and other questions related to integral
quadratic forms
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 to ...
. 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 had already discovered a multiplication of quadratic integers of the same , which allowed them to solve some cases of
Pell's equation
Pell's equation, also called the PellâFermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...
.
The characterization given in of the quadratic integers was first given by
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 â 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
in 1871.
Definition
A quadratic integer is an
algebraic integer
In algebraic number theory, an algebraic integer is a complex number which 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 form ...
, which solves an equation of the form , with ''b'' and ''c''
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (â1, â2, â3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. Each quadratic integer that is not an integer is not
rational
Rationality is the quality of being guided by or based on reasons. 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 abi ...
ânamely, it's a real
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
if and non-real if âand lies in a uniquely determined
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
, the extension of
generated by the square-root of the unique
square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...
that satisfies for some integer . If is positive, the quadratic integer is real. If D < 0, it is ''imaginary'' (that is, complex and nonreal).
The quadratic integers (including the ordinary integers) that belong to a quadratic field
form an
integral domain
In mathematics, specifically abstract algebra, 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 set ...
called the ''ring of integers of''
Although the quadratic integers belonging to a given quadratic field form a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, the set of ''all'' quadratic integers is not a ring because it is not closed under
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
or
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
. 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 two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
where is a
square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...
. 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
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
In other words, every quadratic integer may be written , where andâ are integers, and where is defined by:
:
(as has been supposed square-free the case
is impossible, since it would imply that D would be divisible by the square 4).
Norm and conjugation
A quadratic integer in
may be written
:,
where andâ 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 is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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 of the ring of the integers of
âsee , below.
Quadratic integer rings
Every
square-free integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...
(different from 0 and 1) defines a quadratic integer ring, which is the
integral domain
In mathematics, specifically abstract algebra, 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 set ...
consisting of the
algebraic integer
In algebraic number theory, an algebraic integer is a complex number which 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 deno ...
of ), which is the
integral closure of in
The ring consists of all roots of all equations whose
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
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
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or âŻ⋅âŻ) is . For example, 4 and â4 are square roots of 16, because .
E ...
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 .
The
fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
is not true in many rings of quadratic integers. However, there is a unique factorization for
ideals, which is expressed by the fact that every ring of algebraic integers is a
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
. Being the simplest examples of algebraic integers, quadratic integers are commonly the starting examples of most studies 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 ...
.
The quadratic integer rings divide in two classes depending on the sign of . If , all elements of
are real, and the ring is a ''real quadratic integer ring''. If , the only real elements of
are the ordinary integers, and the ring is a ''complex quadratic integer ring''.
For real quadratic integer rings, the
class number, which measures the failure of unique factorization, is given i
OEIS A003649 for the imaginary case, they are given i
OEIS A000924
Units
A quadratic integer is a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
in the ring of the integers of
if and only if its norm is or . In the first case its
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
is its conjugate. It is the negation of its conjugate in the second case.
If , the ring of the integers of
has at most six units. In the case of 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 /ma ...
s (), the four units are . In the case of the
Eisenstein integers (), the six units are . For all other negative , there are only two units, which are and .
If , the ring of the integers of
has infinitely many units that are equal to , where is an arbitrary integer, and is a particular unit called a ''
fundamental unit''. Given a fundamental unit , there are three other fundamental units, its conjugate
and also
and
Commonly, one calls ''the'' fundamental unit, the unique one which has an absolute value greater than 1 (as a real number). It is the unique fundamental unit that may be written as , with and positive (integers or halves of integers).
The fundamental units for the 10 smallest positive square-free are , , (the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
), , , , , , , . For larger , the coefficients of the fundamental unit may be very large. For example, for , the fundamental units are respectively
, and .
Examples of complex quadratic integer rings
For < 0, Ï is a complex (
imaginary or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic
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 form ...
s.
* A classic example is