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In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
Contents
- History
- Definition
- Norm and conjugation
- Units
- Quadratic integer rings
- Examples of complex quadratic integer rings
- Examples of real quadratic integer rings
- Principal rings of quadratic integers
- Euclidean rings of quadratic integers
- References
with B and C integers. When algebraic integers are considered, the 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, and other questions related to integral quadratic forms. 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.
Definition
A quadratic integer is an algebraic integer of degree two. More explicitly, it is a complex number
The quadratic integers (including the ordinary integers), which belong to a quadratic field
Here and in the following, D is supposed to be a square-free integer. This does not restrict the generality, as the equality √a2D = a√D (for any positive integer a) implies
Every quadratic integer may be written a + ωb , where a and b are integers, and where ω is defined by:
(as D has been supposed square-free the case
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
Norm and conjugation
A quadratic integer in
where either a and b are either both integers, or, only if D ≡ 1 (mod 4), both halves of odd integers. The norm of such a quadratic integer is
N(a + b√D) = a2 – Db2.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√D has a conjugate
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
Units
A quadratic integer is a unit in the ring of the integers of
If D < 0, the ring of the integers of
If D > 0, the ring of the integers of
The fundamental units for the 10 smallest positive square-free D are 1 + √2, 2 + √3, 1 + √5/2 (the golden ratio), 5 + 2√6, 8 + 3√7, 3 + √10, 10 + 3√11, 3 + √13/2, 15 + 4√14, 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
Every square-free integer (different from 0 and 1) D defines a quadratic integer ring, which is the integral domain of the algebraic integers contained in
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
Examples of complex quadratic integer rings
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 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
which is not even a unique factorization domain. This can be shown as follows.
In
The factors 3,
The ideals
Examples of real quadratic integer rings
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 X2 − D Y2 = 1, a Diophantine equation that has been widely studied, are the units of these rings, for D ≡ 2, 3 (mod 4).
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[√−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
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
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 the OEIS).There is no other ring of quadratic integers that is Euclidean with the norm as a Euclidean function.
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. The values D = 14, 69 were the first for which the ring of quadratic integers was proven to be Euclidean, but not norm-Euclidean.