Quadratic integer

In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are the solutions of equations of the form

x 2 + Bx + C

= 0

with $B$ and $C$ integers. They are thus algebraic integers of the degree two. When algebraic integers are considered, usual integers are often called rational integers.

Common examples of quadratic integers are the square roots of 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 $-1 + \sqrt -3 / 2$ , which generates the Eisenstein integers.

Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations. 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 $D$ , which allowed them to solve some cases of Pell's equation.

The characterization of the quadratic integers was first given by Richard Dedekind in 1871.^[1]^[2]

Definition

A quadratic integer is a complex number which is a solution of an equation of the form

x 2 + Bx + C = 0

with B and C integers. In other words, a quadratic integer is an algebraic integer in a quadratic field. Each quadratic integer that is not an integer lies in a uniquely determined quadratic field, namely, the extension of $\mathbb{Q}$ generated by the square-root of $B 2 -4 C$ , which can always be written in the form $\mathbb{Q}(\sqrt{D}),$ where $D$ is the unique square-free integer for which $B 2 - 4 C = DE 2$ for some integer $E$ .

The quadratic integers (including the ordinary integers), which belong to a quadratic fieds $\mathbb{Q}(\sqrt{D}),$ form a integral domain called ring of integers of $\mathbb{Q}(\sqrt{D}).$

Here and in the following, $D$ is supposed to be a square-free integer. This does not restrict the generality, as the equality $\sqrt a 2 D = a \sqrt D$ (for any positive integer $a$ ) implies $\mathbb{Q}(\sqrt{D})=\mathbb{Q}(\sqrt{a^2D}).$

Every quadratic integer may be written $a + ωb$ , where $a$ and $b$ are integers, and where $ω$ 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 $D$ has been supposed square-free the case $D \equiv 0\pmod{4}$ is impossible, since it would imply that D would be divisible by the square 4).

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, as $\sqrt{2}+\sqrt{3}$ is an algebraic integer, which has a minimal polynomial of degree four.

Norm and conjugation

A quadratic integer in $\mathbb{Q}(\sqrt{D})$ may be written

a + b \sqrt D

where either $a$ and $b$ are either integers, or, only if $D \equiv 1 (mod 4)$ , halves of odd integers. The norm of such a quadratic integer is

N (a + b \sqrt D) = a 2 - b 2 D

The norm of a quadratic integer is always an integer. If $D < 0$ , 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}.

An algebraic integer has the same norm as its conjugate, and this norm is the product of the algebraic integer and its conjugate. The conjugate of a sum or a product of algebraic 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 $\mathbb{Q}(\sqrt{D}).$

Units

A quadratic integer is a unit in the ring of the integers of $\mathbb{Q}(\sqrt{D})$ if and only if its norm is $1$ or $-1$ . In the first case its multiplicative inverse is its conjugate. It is the opposite of its conjugate in the second case.

If $D < 0$ , the ring of the integers of $\mathbb{Q}(\sqrt{D})$ has at most six units. In the case of the Gaussian integers ( $D = -1$ ), the four units are $1, -1, \sqrt -1, - \sqrt -1$ . In the case of the Eisenstein integers ( $D = -3$ ), the six units are $\pm1, \pm1 \pm \sqrt -3 / 2$ . For all other negative $D$ , there are only two units that are $1$ and $-1$ .

If $D > 0$ , the ring of the integers of $\mathbb{Q}(\sqrt{D})$ has infinitely many units that are equal to $\pm u i$ , where $i$ is an arbitrary integer, and $u$ is a particular unit called a fundamental unit. Given a fundamental unit $u$ , 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 $a + b \sqrt D$ , with $a$ and $b$ positive (integers or halves of integers).

The fundamental units for the 10 smallest positive square-free $D$ are $1 + \sqrt 2$ , $2 + \sqrt 3$ , $1 + \sqrt 5 / 2$ (the golden ratio), $5 + 2 \sqrt 6$ , $8 + 3 \sqrt 7$ , $3 + \sqrt 10$ , $10 + 3 \sqrt 11$ , $3 + \sqrt 13 / 2$ , $15 + 4 \sqrt 14$ , $4 + \sqrt 15$ . For larger $D$ , the coefficients of the fundamental unit may be very large. For example, for $D = 19, 31, 43$ , the fundamental units are respectively $170 + 39 \sqrt 19$ , $1520 + 273 \sqrt 31$ and $3482 + 531 \sqrt 43$ .

Quadratic integer rings

Every square-free integer (different of 0 and 1) $D$ defines a quadratic integer ring, which is the integral domain of the algebraic integers contained in $\mathbf{Q}(\sqrt{D}).$ It is the set $Z [ω] = a + ωb : a, b \in Z$ , where $ω$ is defined as above. It is called the ring of integers of Q(√D) and often denoted $\mathcal{O}_{\mathbf{Q}(\sqrt{D})}.$ By definition, it is the integral closure of $Z$ in $\mathbf{Q}(\sqrt{D}).$

The properties of the quadratic integers (and more generally of algebraic integers) has been a long standing problem, which has motivated the elaboration of the notions of ring and ideal. In particular 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.

Quadratic integer rings and their associated quadratic fields are thus commonly the starting examples of most studies of algebraic number fields.

The quadratic integer rings divide in two classes depending on the sign of $D$ . If $D > 0$ , all elements of $\mathcal{O}_{\mathbf{Q}(\sqrt{D})}$ are real, and the ring is a real quadratic integer ring. If $D < 0$ , the only real elements of $\mathcal{O}_{\mathbf{Q}(\sqrt{D})}$ are the ordinary integers, and the ring is a complex quadratic integer ring.

Examples of complex quadratic integer rings

Gaussian integers

Eisenstein primes

For $D$ < 0, ω 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 numbers.

A classic example is $\mathbf{Z}[\sqrt{-1}]$ , the Gaussian integers, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law.^[3]
The elements in $\mathcal{O}_{\mathbf{Q}(\sqrt{-3})} = \mathbf{Z}\left[{{1 + \sqrt{-3}} \over 2}\right]$ are called Eisenstein integers.

Both rings mentioned above are rings of integers of cyclotomic fields Q(ζ₄) and Q(ζ₃) correspondingly. In contrast, Z[√−3] is not even a Dedekind domain.

Both 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 $\pm1$ 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.

Examples of real quadratic integer rings

Powers of the golden ratio

For $D > 0$ , $ω$ is a positive irrational real number, and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the Pell's equation $X 2 - D Y 2 = 1$ , a Diophantine equation that has been widely studied, are the units of these rings, for $D \equiv 2, 3 (mod 4)$ .

For $D = 5$ , $ω = 1+ \sqrt 5 / 2$ is the golden ratio. This ring was studied by Peter Gustav Lejeune Dirichlet. Its invertible elements have the form $\pmω n$ , where $n$ is an arbitrary integer. This ring also arises from studying 5-fold rotational symmetry on Euclidean plane, for example, Penrose tilings.^[4]
Indian mathematician Brahmagupta treated the Pell's equation $X 2 - 61 Y 2 = 1$ , corresponding to the ring is $Z [\sqrt 61]$ . Some results were presented to European community by Pierre Fermat in 1657.

Principal rings of quadratic integers

Unique factorization property is not always verified for rings of quadratic integers, as seen above for the case of $Z [\sqrt -5]$ . However, as for every Dedekind domain, a ring of quadratic integers is a unique factorization domain if and only if it is a principal ideal domain. This occurs if and only if the class number of the corresponding quadratic field is one.

The imaginary rings of quadratic integers that are principal ideal rings have been completely determined. These are $\mathcal{O}_{\mathbf{Q}(\sqrt{D})}$ for

D = -1, -2, -3, -7, -11, -19, -43, -67, -163

This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark 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 $D > 0$ , 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 if it is a Euclidean domain. This problem has been completely solved as follows.

Equipped with the norm $N(a + b\sqrt{D}) = a^2 - Db^2,$ as an Euclidean function, $\mathcal{O}_{\mathbf{Q}(\sqrt{D})}$ is an Euclidean domain for negative $D$ when

D = -1, -2, -3, -7, -11

, ^[5]

and, for positive $D$ , when

D = 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73

(sequence A048981 in OEIS).

There is no other ring of quadratic integers that is Euclidean with the norm as a Euclidean function.^[6]

For negative $D$ , a ring of quadratic integers is Euclidean if and only if the norm is a Euclidean function for it. It follows that, for

D = -19, -43, -67, -163

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.^[7] The values D = 14, 69 were the first for which the ring of quadratic integers was proven to be Euclidean, but not norm-Euclidean.^[8]^[9]

Notes

↑ Dedekind 1871, Supplement X, p. 447
↑ Bourbaki 1994, p. 99
↑ Dummit, pg. 229
↑ de Bruijn, N. G. (1981), "Algebraic theory of Penrose's non-periodic tilings of the plane, I, II" (PDF), Indagationes mathematicae 43 (1): 39–66
↑ Dummit, pg. 272
↑ LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. pp. II:57,81. ISBN 978-0-486-42539-9. Zbl 1009.11001.
↑ P. Weinberger, On Euclidean rings of algebraic integers. In: Analytic Number Theory (St. Louis, 1972), Proc. Sympos. Pure Math. 24(1973), 321–332.
↑ M. Harper, $\Bbb {Z}[\sqrt{14}]$ is Euclidean. Canad. J. Math. 56(2004), 55–70.
↑ David A. Clark, A quadratic field which is Euclidean but not norm-Euclidean, Manuscripta Mathematica, 83(1994), 327–330

References

Bourbaki, Nicolas (1994), Elements of the history of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-64767-6, MR 1290116 . Translated from the original French by John Meldrum
Dedekind, Richard (1871), Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet (2 ed.), Vieweg . Retrieved 5. August 2009
Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed.
J.S. Milne. Algebraic Number Theory, Version 3.01, September 28, 2008. online lecture note

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