For the real quadratic field K=\mathbf{Q}(\sqrt{d}) (with
d square-free), the fundamental unit ε is commonly normalized so that (as a real number). Then it is uniquely characterized as the minimal unit among those that are greater than 1. If Δ denotes the
discriminant of
K, then the fundamental unit is :\varepsilon=\frac{a+b\sqrt{\Delta}}{2} where (
a,
b) is the smallest solution to :x^2-\Delta y^2=\pm4 in positive integers. This equation is basically
Pell's equation or the negative Pell equation and its solutions can be obtained similarly using the
continued fraction expansion of \sqrt{\Delta}. Whether or not
x2 − Δ
y2 = −4 has a solution determines whether or not the
class group of
K is the same as its
narrow class group, or equivalently, whether or not there is a unit of norm −1 in
K. This equation is known to have a solution if, and only if, the period of the continued fraction expansion of \sqrt{\Delta} is odd. A simpler relation can be obtained using congruences: if Δ is divisible by a prime that is congruent to 3 modulo 4, then
K does not have a unit of norm −1. However, the converse does not hold as shown by the example
d = 34. In the early 1990s, Peter Stevenhagen proposed a probabilistic model that led him to a conjecture on how often the converse fails. Specifically, if
D(
X) is the number of real quadratic fields whose discriminant Δ −(
X) is those who have a unit of norm −1, then :\lim_{X\rightarrow\infty}\frac{D^-(X)}{D(X)}=1-\prod_{j\geq1\text{ odd}}\left(1-2^{-j}\right). In other words, the converse fails about 42% of the time. As of March 2012, a recent result towards this conjecture was provided by
Étienne Fouvry and Jürgen Klüners who show that the converse fails between 33% and 59% of the time. In 2022, Peter Koymans and Carlo Pagano claimed a complete proof of Stevenhagen's conjecture. ==Cubic fields==