Quadratic integer

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In number theory, quadratic integers are a generalization of the rational integers to quadratic fields. These are algebraic integers of the degree 2. Important examples include the Gaussian integers and the Eisenstein integers. Though they have been studied for more than a hundred years, many open problems remain.

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).

This characterization[clarification needed] was first given by Richard Dedekind in 1871.[1][2]

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.

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 study of quadratic integers is related to the study of quadratic forms with integer coefficients.

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} . It is called the 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 Dedekind domain.

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

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]