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 ${\displaystyle K=\mathbf {Q} ({\sqrt {d}})}$ (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

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

where (a, b) is the smallest solution to ^[2]

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

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 ${\displaystyle {\sqrt {\Delta }}}$.

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 ${\displaystyle {\sqrt {\Delta }}}$ 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]

${\displaystyle \lim _{X\rightarrow \infty }{\frac {D^{-}(X)}{D(X)}}=1-\prod _{j\geq 1{\text{ odd}}}\left(1-2^{-j}\right).}$

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]

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

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

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

so ${\displaystyle \epsilon ^{3}\approx 56.9>20.25}$.

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. Koymans, Peter; Pagano, Carlo (2022-01-31). "On Stevenhagen's conjecture". arXiv: 2201.13424 [math.NT].
7. Alaca & Williams 2004 , Theorem 13.6.1

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
• Fouvry, Étienne; Klüners, Jürgen (2010), "On the negative Pell equation", Annals of Mathematics , 2 (3): 2035–2104, doi: 10.4007/annals.2010.172.2035 , MR   2726105
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, ISBN   978-3-540-65399-8, MR   1697859, Zbl   0956.11021
• Stevenhagen, Peter (1993), "The number of real quadratic fields having units of negative norm", Experimental Mathematics, 2 (2): 121–136, CiteSeerX   10.1.1.27.3512 , doi:10.1080/10586458.1993.10504272, MR   1259426

External links

• Weisstein, Eric W. "Fundamental Unit". MathWorld .