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. 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 [math]\displaystyle{ K=\mathbf{Q}(\sqrt{d}) }[/math] (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

[math]\displaystyle{ \varepsilon=\frac{a+b\sqrt{\Delta}}{2} }[/math]

where (a, b) is the smallest solution to

[math]\displaystyle{ x^2-\Delta y^2=\pm4 }[/math]

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 [math]\displaystyle{ \sqrt{\Delta} }[/math].

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 [math]\displaystyle{ \sqrt{\Delta} }[/math] 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 not divisible by a prime congruent to 3 modulo 4 and D⁻(X) is those who have a unit of norm −1, then

[math]\displaystyle{ \lim_{X\rightarrow\infty}\frac{D^-(X)}{D(X)}=1-\prod_{j\geq1\text{ odd}}\left(1-2^{-j}\right). }[/math]

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

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

[math]\displaystyle{ \epsilon^3\gt \frac{|\Delta|-27}{4}. }[/math]

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

[math]\displaystyle{ \frac{|\Delta|-27}{4}=20.25 }[/math]

so [math]\displaystyle{ \epsilon^3 \approx 56.9 \gt 20.25 }[/math].

References

• Alaca, Şaban; Williams, Kenneth S. (2004), Introductory algebraic number theory, Cambridge University Press, ISBN 978-0-521-54011-7 
• Duncan Buell (1989), Binary quadratic forms: classical theory and modern computations, Springer-Verlag, pp. 92–93, ISBN 978-0-387-97037-0, https://archive.org/details/binaryquadraticf0000buel/page/92 
• Fouvry, Étienne; Klüners, Jürgen (2010), "On the negative Pell equation", Annals of Mathematics 2 (3): 2035–2104, doi:10.4007/annals.2010.172.2035 
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8 
• Stevenhagen, Peter (1993), "The number of real quadratic fields having units of negative norm", Experimental Mathematics 2 (2): 121–136, doi:10.1080/10586458.1993.10504272 

External links

• Weisstein, Eric W.. "Fundamental Unit". http://mathworld.wolfram.com/FundamentalUnit.html.