is a square
, then Q
)), the set of all numbers a + b
and b rational
), is called a quadratic field
. If d
is positive, it is called a real
quadratic field; if d
is negative, then it is called an imaginary
quadratic field. A quadratic field is, in fact, a field
The defining equation of z in Q(sqrt(d)) is the quadratic polynomial with integer coefficients of which z is a root. If z = (a + b sqrt(d)) / c, then the defining equation for z would be (c x - a - b sqrt(d))(c x - a +b sqrt(d)) = c^2 x^2 - 2 a c x + a^2 - b^2 d.
We can then define a quadratic integer as a number in Q(sqrt(d)) such that its defining equation is monic, i.e, that its leading coefficient is 1. Notice that this definition is compatible with the established definition of Z, the integers (in this context, Z is referred to as the rational integers to avoid confusion). The integers of Q(sqrt(-1)) are generally called the Gaussian integers.
If d is equal to 2 or 3 (mod 4) then the quadratic integers look like a + b sqrt(d), where a and b are rational integers. If d is equal to 1 (mod 4), then the quadratic integers look like a + b[(1 + sqrt(d)) / 2], again with a and b rational integers.
Unlike Q, Q(sqrt(d)) is not always a unique factorization domain; we don't always get to factor nicely. For example, Q(sqrt(-10)) is not a unique factorization domain, since 6 = 2 x 3 = (4 + sqrt(10))(4 - sqrt(10)). The question of which values of d give you unique factorization domain has been a matter of long debate. It is still unknown whether there are infinitely many such d. It is, however, known in the complex case:
Theorem: If d < 0, then Q(sqrt(d)) is a unique factorization domain if and only if d = -1, -2, -3, -7, -11, -19, -43, -67, or -163.
This theorem was originally conjectured by Gauss in Disquisitiones Arithmeticae. In 1934, it was shown there could be at most 10; and the nine above were already known. But it wasn't until 1966 that Q(sqrt(d)) was shown not to be a UFD for all d < -163.