Quadratic integer

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

x2 + 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 2, and the complex number i = –1, which generates the Gaussian integers. Another common example is the non-real cubic root of unity 1 + –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[edit]

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.[citation needed]

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

Definition[edit]

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

x2 + Bx + C = 0

with B and C integers. In other words, a quadratic integer is an algebraic integer, which belongs to a quadratic field. The quadratic field to which belongs a quadratic integer that is not an integer, and satisfies above quadratic equation is \mathbb{Q}(\sqrt{D}), where D is the unique square-free integer such that B2 – 4C = DE2 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}). They 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}

(D is a square-free integer, and the case  D \equiv 0\pmod{4} is impossible, since it would imply that D would be divisible by 4, a perfect square, which contradicts the fact that D is square-free).

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[edit]

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

a + bD,

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

N(a + bD) = a2b2D.

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 + bD 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 it 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[edit]

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, –1, ––1. In the case of the Eisenstein integers (D = –3), the six units are ±1, ±1 ± –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 ±ui, 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 + bD, with a and b positive (integers or halves of integers).

The fundamental units for the 10 smallest positive square-free D are 1 + 2, 2 + 3, 1 + 5/2 (the golden ratio), 5 + 26, 8 + 37, 3 + 10, 10 + 311, 3 + 13/2, 15 + 414, 4 + 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 19, 1520 + 273 31 and 3482 + 531 43.

Quadratic integer rings[edit]

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, bZ, 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}), and, as such it is a Dedekind domain.

Quadratic integer rings and their associated quadratic fields are 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[edit]

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.

Both rings mentioned above are rings of integers of cyclotomic fields Q4) and Q3) correspondingly. In contrast, Z[−3] is not even a Dedekind domain.

Examples of real quadratic integer rings[edit]

For D > 0, ω is a positive irrational and the corresponding quadratic integer ring is a set of algebraic real numbers. Pell's equation X2DY2 = 1, a case of Diophantine equations, naturally leads to these rings for D ≡ 2, 3 (mod 4) . Algebraic study of real quadratic integer rings involves determining of the invertible elements group.

Powers of the golden ratio
  • For D = 5, ω is the golden ratio. A non-negative real number belongs to the ring Z[(1+5)/2] if and only if it can be encoded in golden ratio base with finite number of 1's. This ring was studied by Peter Gustav Lejeune Dirichlet. Its invertible elements have the form ±ωn, 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 equation X261Y2 = 1, where the corresponding ring is Z[61]. Some results were presented to European community by Pierre Fermat in 1657.

Class number[edit]

Equipped with the norm

N(a + b\sqrt{D}) = a^2 - Db^2,

\mathcal{O}_{\mathbf{Q}(\sqrt{D})} is an Euclidean domain (and thus a unique factorization domain, UFD) for negative D when D = −1, −2, −3, −7, −11.[5] On the other hand, it turned out that Z[−5] is not a UFD because, for example, 6 has two distinct factorizations into irreducibles:

6 = 2(3) = (1 + \sqrt{-5}) (1 - \sqrt{-5}).

(In fact, Z[−5] has class number 2.[6]) The failure of the unique factorization led Ernst Kummer and Dedekind to develop a theory that would enlarge the set of "prime numbers"; the result was the introduction of the notion of ideal, and the definition of what is now called a Dedekind domain: All the ring of integers of number fields are Dedekind domains, and the ideals of a Dedekind domain have the property of unique factorization into products of prime ideals.

Being a Dedekind domain, a quadratic integer ring is a UFD if and only if it is a principal ideal domain (i.e., its class number is one). However, there are quadratic integer rings that are principal ideal domains but not Euclidean domains. For example, Q[−19] has class number 1 but its ring of integers is not Euclidean.[6] There are effective methods to compute ideal class groups of quadratic integer rings, but many theoretical questions about their structure are still open after a hundred years.[when?]

Notes[edit]

  1. ^ Dedekind 1871, Supplement X, p. 447
  2. ^ Bourbaki 1994, p. 99
  3. ^ Dummit, pg. 229
  4. ^ 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 
  5. ^ Dummit, pg. 272
  6. ^ a b Milne, pg. 64

References[edit]