Quadratic integer

In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree two whose coefficients are integers, i.e. quadratic integers are algebraic integers 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 roots of rational integers, such as \sqrt{2} , and the complex number i=\sqrt{-1} , which generates the Gaussian integers. Another common example is the non-real cubic root of unity \frac{-1+\sqrt{-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.

In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree two whose coefficients are integers, i.e. quadratic integers are algebraic integers 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 roots of rational integers, such as \sqrt{2} , and the complex number i=\sqrt{-1} , which generates the Gaussian integers. Another common example is the non-real cubic root of unity \frac{-1+\sqrt{-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. A quadratic integer is an algebraic integer of degree two. More explicitly, it is a complex number x = (-b\pm\sqrt{b^2-4c})/2 , which solves an equation of the form , with and integers. Each quadratic integer that is not an integer is not rational—namely, it is a real irrational number if and non-real if —and lies in a uniquely determined quadratic field \mathbb{Q}(\sqrt{D}\,) , the extension of \mathbb{Q} 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{Q}(\sqrt{D}\,) form an integral domain called the ring of integers of \mathbb{Q}(\sqrt{D}\,). 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{2} and \sqrt{3} are quadratic integers, but 1+\sqrt{2}+\sqrt{3} and (1+\sqrt{2})\cdot\sqrt{3} are not, as their minimal polynomials have degree four. Here and in the following, the quadratic integers that are considered belong to a quadratic field \mathbb{Q}(\sqrt{D}\,), where is a square-free integer. This does not restrict the generality, as the equality \sqrt{a^2D} = a\sqrt{D} (for any positive integer ) implies \mathbb{Q}(\sqrt{D}\,) = \mathbb{Q}(\sqrt{a^2D}\,). An element of \mathbb{Q}(\sqrt{D}\,) 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{a}{2}+\frac{b}{2}\sqrt{D}, with and both odd. In other words, every quadratic integer may be written , where and are integers, and where is defined by \omega = \begin{cases} \sqrt{D} & \mbox{if }D \equiv 2, 3 \pmod{4} \\ {{1 + \sqrt{D}} \over 2} & \mbox{if }D \equiv 1 \pmod{4} \end{cases} (as has been supposed square-free the case D \equiv 0\pmod{4} is impossible, since it would imply that is divisible by the square 4). a+b\sqrt{D} , 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{D})=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 as a complex number (this is false if D > 0 ). 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 a+b\sqrt{D} has a conjugate \overline{a+b\sqrt{D}} = a-b\sqrt{D}. 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 \mathbb{Q}(\sqrt{D}\,) – see , below. 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 \mathbb{Q}(\sqrt{D}\,). It is the set \mathbb{Z}[\omega] = \{ a + \omega b : a,b \in \mathbb{Z} \} where \omega = \tfrac{1+\sqrt D}{2} if , and otherwise. It is often denoted \mathcal{O}_{\mathbb{Q}(\sqrt{D}\,)} , because it is the ring of integers of \mathbb{Q}(\sqrt{D}\,) , which is the integral closure of \mathbb{Z} in \mathbb{Q}(\sqrt{D}\,). The ring \mathbb{Z}[\omega] consists of all roots of all equations whose discriminant is the product of by the square of an integer. In particular belongs to \mathbb{Z}[\omega] , 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 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. The quadratic integer rings divide in two classes depending on the sign of . If , all elements of \mathcal{O}_{\mathbb{Q}(\sqrt{D}\,)} are real, and the ring is a real quadratic integer ring. If , the only real elements of \mathcal{O}_{\mathbb{Q}(\sqrt{D}\,)} 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 in OEIS A003649; for the imaginary case, they are given in OEIS A000924. If , the ring of the integers of \mathbb{Q}(\sqrt{D}\,) has at most six units. In the case of the Gaussian integers ( ), the four units are 1, -1, \sqrt{-1}, -\sqrt{-1} . In the case of the Eisenstein integers ( ), the six units are \pm 1, \frac{\pm 1 \pm \sqrt{-3}}{2} . For all other negative , there are only two units, which are and . If , the ring of the integers of \mathbb{Q}(\sqrt{D}\,) 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{u}, and also -u and -\overline{u}. 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} (the silver ratio), 2+\sqrt{3} , \frac{1+\sqrt{5}}{2} (the golden ratio), 5+2\sqrt{6} , 8+3\sqrt{7} , 3+\sqrt{10} , 10+3\sqrt{11} , \frac{3+\sqrt{13}}{2} , 15+4\sqrt{14} , 4+\sqrt{15} . For larger , the coefficients of the fundamental unit may be very large. For example, for , the fundamental units are respectively 170+39\sqrt{19} , 1520+273\sqrt{31} and 3482+531\sqrt{43} . The elements in \mathcal{O}_{\mathbf{Q}(\sqrt{-3}\,)} = \mathbf{Z}\left[\tfrac12\bigl(1 + \sqrt{-3}~\!\bigr)\right] are called Eisenstein integers. The elements in \mathcal{O}_{\mathbf{Q}(\sqrt{-7}\,)} = \mathbf{Z}\left[\tfrac12\bigl(1 + \sqrt{-7}~\!\bigr)\right] are called Kleinian integers The first two rings mentioned above are rings of integers of cyclotomic fields Q(ζ 4 ) and Q(ζ 3 ) correspondingly. In contrast, \mathbf{Z}\bigl[\sqrt{-3}~\!\bigr] is not even a Dedekind domain. All the above examples are principal ideal rings and also Euclidean domains for the norm. This is not the case for \mathcal{O}_{\mathbf{Q}(\sqrt{-5}\,)} = \mathbf{Z}\left[\sqrt{-5}\,\right], which is not even a unique factorization domain. This can be shown as follows. In \mathcal{O}_{\mathbf{Q}(\sqrt{-5}\,)}, we have 9 = 3\cdot3 = (2+\sqrt{-5})(2-\sqrt{-5}). The factors 3, 2+\sqrt{-5} and 2-\sqrt{-5} 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{-5}\,\rangle and \langle 3, 1-\sqrt{-5}\,\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. N(a + b\sqrt{D}\,) = |a^2 - Db^2| as a Euclidean function, \mathcal{O}_{\mathbf{Q}(\sqrt{D}\,)} 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 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. The values were the first for which the ring of quadratic integers was proven to be Euclidean, but not norm-Euclidean.

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