Quadratic integer

Another common example is the non-real cubic root of unity

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.

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 given in § Explicit representation of the quadratic integers was first given by Richard Dedekind in 1871.

, which solves an equation of the form x2 + bx + c = 0, with b and c integers.

Each quadratic integer that is not an integer is not rational – namely, it's a real irrational number if b2 − 4c > 0 and non-real if b2 − 4c < 0 – and lies in a uniquely determined quadratic field

form an integral domain called the ring of integers of

If D < 0, the norm of a quadratic integer is the square of its absolute value as a complex number (this is false if

This means that the conjugation is an automorphism of the ring of the integers of

consists of all roots of all equations x2 + Bx + C = 0 whose discriminant B2 − 4C is the product of D by the square of an integer.

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.

[4] The quadratic integer rings divide in two classes depending on the sign of D. If D > 0, all elements of

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.

Commonly, one calls "the fundamental unit" the unique one which has an absolute value greater than 1 (as a real number).

The fundamental units for the 10 smallest positive square-free D are

For larger D, the coefficients of the fundamental unit may be very large.

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 above examples are principal ideal rings and also Euclidean domains for the norm.

For D > 0, ω is a positive irrational real number, and the corresponding quadratic integer ring is a set of algebraic real numbers.

The unique factorization property is not always verified for rings of quadratic integers, as seen above for the case of Z[√−5].

This occurs if and only if the class number of the corresponding quadratic field is one.

for 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.

However, the complete list is not known; it is not even known if the number of these principal ideal rings is finite or not.

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.

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