Fundamental unit (number theory)

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). Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units.^[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).

Real quadratic fields

For the real quadratic field $K=\mathbf {Q} ({\sqrt {d}})$ (with d square-free), the fundamental unit ε is commonly normalized so that $ε > 1$ (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^[2]

x^{2}-\Delta y^{2}=\pm 4

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 − 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 ${\sqrt {\Delta }}$ 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.^[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]

\lim _{X\rightarrow \infty }{\frac {D^{-}(X)}{D(X)}}=1-\prod _{j\geq 1{\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^[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.

Cubic fields

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. If the discriminant Δ of K satisfies |Δ| ≥ 33, then^[7]

\epsilon ^{3}>{\frac {|\Delta |-27}{4}}.

For example, the fundamental unit of $\mathbf {Q} ({\sqrt[{3}]{2}})$ is $\epsilon =1+{\sqrt[{3}]{2}}+{\sqrt[{3}]{2^{2}}},$ and $\epsilon ^{3}\approx 56.9$ whereas the discriminant of this field is −108 thus