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. ).

Real quadratic fields

For the real quadratic field $K=\mathbf(\sqrt)$ (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
:$\varepsilon=\frac$
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 $\sqrt$.

Whether or not ''x''² − Δ''y''² = −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 $\sqrt$ 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
:$\lim_\frac=1-\prod_\left(1-2^\right).$
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.

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
:$\epsilon^3>\frac.$
For example, the fundamental unit of \mathbf(\sqrt is \epsilon = 1+\sqrt \sqrt and $\epsilon^3\approx 56.9$ whereas the discriminant of this field is −108 thus
:$\frac=20.25$
so $\epsilon^3 \approx 56.9 > 20.25$.

Notes

References

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External links

* {{MathWorld, title=Fundamental Unit, urlname=FundamentalUnit

Algebraic number theory