In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic).
[1] Some authors use the term fundamental unit to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. Neukirch 1999, p. 42).
If Δ denotes the discriminant of K, then the fundamental unit is where (a, b) is the smallest solution to[2] in positive integers.
Whether or not x2 − Δy2 = −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
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.
[3] 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 not divisible by a prime congruent to 3 modulo 4 and D−(X) is those who have a unit of norm −1, then[4] 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[5] who show that the converse fails between 33% and 59% of the time.
In 2022, Peter Koymans and Carlo Pagano[6] claimed a complete proof of Stevenhagen's conjecture.
If K is a complex cubic field then it has a unique real embedding and the fundamental unit ε can be picked uniquely such that |ε| > 1 in this embedding.