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 (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

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

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 .

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 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]}

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]}

For example, the fundamental unit of is and whereas the discriminant of this field is −108 and

so .

## Notes

**^**Alaca & Williams 2004, §13.4**^**Neukirch 1999, Exercise I.7.1**^**Alaca & Williams 2004, Table 11.5.4**^**Stevenhagen 1993, Conjecture 1.4**^**Fouvry & Klüners 2010**^**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, J��rgen (1999),
*Algebraic Number Theory*,*Grundlehren der mathematischen Wissenschaften*,**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