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.{{harvnb|Alaca|Williams|2004|loc=§13.4}} 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. {{harvnb|Neukirch|1999|loc=p. 42}}).

Real quadratic fields

For the real quadratic field $K=\mathbf{Q}(\sqrt{d})$ (with d square-free), the fundamental unit ε is commonly normalized so that {{math|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

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

where (a, b) is the smallest solution to{{harvnb|Neukirch|1999|loc=Exercise I.7.1}}

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

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.{{harvnb|Alaca|Williams|2004|loc=Table 11.5.4}} 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{{harvnb|Stevenhagen|1993|loc=Conjecture 1.4}}

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

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{{harvnb|Fouvry|Klüners|2010}} who show that the converse fails between 33% and 59% of the time. In 2022, Peter Koymans and Carlo Pagano{{cite arXiv |last1=Koymans |first1=Peter |last2=Pagano |first2=Carlo |date=2022-01-31 |title=On Stevenhagen's conjecture |class=math.NT |eprint=2201.13424 }} 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{{harvnb|Alaca|Williams|2004|loc=Theorem 13.6.1}}

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

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

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

so $\epsilon^3 \approx 56.9 > 20.25$ .

Notes

References

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| first=Şaban

| last2=Williams

| first2=Kenneth S.

| title=Introductory algebraic number theory

| year=2004

| publisher=Cambridge University Press

| isbn=978-0-521-54011-7

}}

{{citation | author=Duncan Buell | title=Binary quadratic forms: classical theory and modern computations | publisher=Springer-Verlag | year=1989 | isbn=978-0-387-97037-0 | pages=[https://archive.org/details/binaryquadraticf0000buel/page/92 92–93] | url=https://archive.org/details/binaryquadraticf0000buel/page/92 }}
{{citation

| last=Fouvry

| first=Étienne

| last2=Klüners

| first2=Jürgen

| title=On the negative Pell equation

| year=2010

| journal=Annals of Mathematics

| volume=2

| number=3

| pages=2035–2104

| mr=2726105

| doi=10.4007/annals.2010.172.2035

| doi-access=free

}}

{{citation

| last = Neukirch

| first = Jürgen

| author-link = Jürgen Neukirch

| title = Algebraic Number Theory

| publisher = Springer-Verlag

| location = Berlin

| series = {{lang|de|Grundlehren der mathematischen Wissenschaften}}

| isbn = 978-3-540-65399-8

| mr = 1697859

| zbl = 0956.11021

| year = 1999

| volume = 322

}}

{{citation

| last=Stevenhagen

| first=Peter

| title=The number of real quadratic fields having units of negative norm

| year=1993

| journal=Experimental Mathematics

| volume=2

| number=2

| pages=121–136

| mr=1259426

| doi=10.1080/10586458.1993.10504272

| citeseerx=10.1.1.27.3512

}}

External links

{{MathWorld|title=Fundamental Unit|urlname=FundamentalUnit}}

Category:Algebraic number theory