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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 \sqrt, 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 ...
i=\sqrt, 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 ...
\frac, 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 ...
x = (-b\pm\sqrt)/2, 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 ...
\mathbb(\sqrt\,), the extension of \mathbb 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 \mathbb(\sqrt\,) 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'' \mathbb(\sqrt\,). 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+\sqrt and \sqrt are quadratic integers, but 1+\sqrt+\sqrt and (1+\sqrt)\cdot\sqrt 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 ...
\mathbb(\sqrt\,), where is a square-free integer. This does not restrict the generality, as the equality \sqrt = a\sqrt(for any positive integer ) implies \mathbb(\sqrt\,) = \mathbb(\sqrt\,). An element of \mathbb(\sqrt\,) is a quadratic integer if and only if there are two integers and such that either : x = a+b\sqrt D, or, if is a multiple of : x = \frac+\frac\sqrt, with and both odd. In other words, every quadratic integer may be written , where and  are integers, and where is defined by : \omega = \begin \sqrt & \mboxD \equiv 2, 3 \pmod \\ & \mboxD \equiv 1 \pmod \end (as has been supposed square-free the case D \equiv 0\pmod 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 \mathbb(\sqrt\,) may be written : a+b\sqrt, where and  are either both integers, or, only if , both halves of odd integers. The norm of such a quadratic integer is : N(a+b\sqrt)=a^2-Db^2. 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 D > 0). 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 a+b\sqrt has a conjugate : \overline = a-b\sqrt. 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 \mathbb(\sqrt\,) – 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 \mathbb(\sqrt\,). It is the set \mathbb
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
= \ where \omega = \tfrac if , and otherwise. It is often denoted \mathcal_, 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 \mathbb(\sqrt\,), 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 \mathbb in \mathbb(\sqrt\,). The ring \mathbb
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
/math> consists of all
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...
of all equations whose discriminant is the product of by the square of an integer. In particular belongs to \mathbb
omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
/math>, 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 . 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 is prime or can be represented uniquely as a product of prime numbers, ...
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. 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 \mathcal_ are real, and the ring is a ''real quadratic integer ring''. If , the only real elements of \mathcal_ 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 in the ring of the integers of \mathbb(\sqrt\,) 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 ra ...
is its conjugate. It is the
negation In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...
of its conjugate in the second case. If , the ring of the integers of \mathbb(\sqrt\,) 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 ...
s (), the four units are 1, -1, \sqrt, -\sqrt. In the case of 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 (), the six units are \pm 1, \frac. For all other negative , there are only two units, which are and . If , the ring of the integers of \mathbb(\sqrt\,) 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 \overline, and also -u and -\overline. 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 1+\sqrt, 2+\sqrt, \frac (the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
), 5+2\sqrt, 8+3\sqrt, 3+\sqrt, 10+3\sqrt, \frac, 15+4\sqrt, 4+\sqrt. For larger , the
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s of the fundamental unit may be very large. For example, for , the fundamental units are respectively 170+39\sqrt, 1520+273\sqrt and 3482+531\sqrt.


Examples of complex quadratic integer rings

For  < 0, 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 for ...
s. * A classic example is \mathbf sqrt\,/math>, 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, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law. * The elements in \mathcal_ = \mathbf\left
right Rights are law, legal, social, or ethics, ethical principles of freedom or Entitlement (fair division), entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal sy ...
/math> are called
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. * The elements in \mathcal_ = \mathbf\left
right Rights are law, legal, social, or ethics, ethical principles of freedom or Entitlement (fair division), entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal sy ...
/math> are called Kleinian integers The first two rings mentioned above are rings of integers of
cyclotomic field In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...
s Q(''ζ''4) and Q(''ζ''3) correspondingly. In contrast, Z[] is not even a Dedekind domain. All the above examples are principal ideal rings and also
Euclidean domain In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of Euclidean division of integers. Th ...
s for the norm. This is not the case for : \mathcal_ = \mathbf\left sqrt\,\right which is not even a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
. This can be shown as follows. In \mathcal_, we have : 9 = 3\cdot3 = (2+\sqrt)(2-\sqrt). The factors 3, 2+\sqrt and 2-\sqrt are irreducible, as they have all a norm of 9, and if they were not irreducible, they would have a factor of norm 3, which is impossible, the norm of an element different of being at least 4. Thus the factorization of 9 into irreducible factors is not unique. The ideals \langle 3, 1+\sqrt\,\rangle and \langle 3, 1-\sqrt\,\rangle are not principal, as a simple computation shows that their product is the ideal generated by 3, and, if they were principal, this would imply that 3 would not be irreducible.


Examples of real quadratic integer rings

For , is a positive irrational
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
, and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the Pell's equation , a
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 ...
that has been widely studied, are the units of these rings, for . * For , is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
. This ring was studied by
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory o ...
. Its units have the form , where is an arbitrary integer. This ring also arises from studying 5-fold
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
on Euclidean plane, for example, Penrose tilings. * Indian mathematician
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...
treated the Pell's equation , corresponding to the ring is . Some results were presented to European community by Pierre Fermat in 1657.


Principal rings of quadratic integers

The unique factorization property is not always verified for rings of quadratic integers, as seen above for the case of . However, as for every Dedekind domain, a ring of quadratic integers is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
if and only if it is a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...
. This occurs if and only if the class number of the corresponding
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 ...
is one. The imaginary rings of quadratic integers that are principal ideal rings have been completely determined. These are \mathcal_ for : . This result was first
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
d by
Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
and proven by Kurt Heegner, although Heegner's proof was not believed until
Harold Stark Harold Mead Stark (born August 6, 1939) is an Americans, American mathematician, specializing in number theory. He is best known for his solution of the Carl Friedrich Gauss, Gauss class number 1 problem, in effect Stark–Heegner theorem, corre ...
gave a later proof in 1967 (see '' Stark–Heegner theorem''). This is a special case of the famous class number problem. There are many known positive integers , for which the ring of quadratic integers is a principal ideal ring. However, the complete list is not known; it is not even known if the number of these principal ideal rings is finite or not.


Euclidean rings of quadratic integers

When a ring of quadratic integers is a principal ideal domain, it is interesting to know whether it is a
Euclidean domain In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of Euclidean division of integers. Th ...
. This problem has been completely solved as follows. Equipped with the norm N(a + b\sqrt\,) = , a^2 - Db^2, as a Euclidean function, \mathcal_ is a Euclidean domain for negative when : , and, for positive , when : . There is no other ring of quadratic integers that is Euclidean with the norm as a Euclidean function. For negative , a ring of quadratic integers is Euclidean if and only if the norm is a Euclidean function for it. It follows that, for : , the four corresponding rings of quadratic integers are among the rare known examples of principal ideal domains that are not Euclidean domains. On the other hand, the
generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whi ...
implies that a ring of ''real'' quadratic integers that is a principal ideal domain is also a Euclidean domain for some Euclidean function, which can indeed differ from the usual norm.P. Weinberger, ''On Euclidean rings of algebraic integers''. In: Analytic Number Theory (St. Louis, 1972), Proc. Sympos. Pure Math. 24(1973), 321–332. The values were the first for which the ring of quadratic integers was proven to be Euclidean, but not norm-Euclidean.


Notes


References

* * * * * * * *


Further reading

* J.S. Milne.
Algebraic Number Theory
', Version 3.01, September 28, 2008. online lecture notes {{refend Algebraic number theory Ring theory Integers