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).

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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. Then it is uniquely characterized as the minimal unit whose absolute value is > 1. If Δ denotes the discriminant of K, then the fundamental unit is

$\epsilon=\frac{a+b\sqrt{\Delta}}{2}$

where (ab) is the smallest solution to[2]

$x^2-\Delta y^2=\pm4$

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 − 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]

$\lim_{X\rightarrow\infty}\frac{D^-(x)}{D(x)}=1-\prod_{j\geq1\text{ odd}}\left(1-2^{-j}\right).$

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[5] 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[6]

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

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

$\frac{|\Delta|-27}{4}=20.25.$

Notes

1. ^ Alaca & Williams 2004, §13.4
2. ^ Neukirch 1999, Exercise I.7.1
3. ^ Alaca & Williams 2004, Table 11.5.4
4. ^ Stevenhagen 1993, Conjecture 1.4
5. ^ Fouvry & Klüners 2010
6. ^ Alaca & Williams 2004, Theorem 13.6.1