View source for Fundamental unit (number theory)

In [[algebraic number theory]], a '''fundamental unit''' is a generator (modulo the [[root of unity|roots of unity]]) for the [[unit group]] of the [[Algebraic integer|ring of integers]] of a [[number field]], when that group has [[rank of an abelian group|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 number field|totally imaginary]] [[quartic field]]. When the unit group has rank ≥&nbsp;1, a basis of it modulo its torsion is called a '''fundamental system of units'''.<ref>{{harvnb|Alaca|Williams|2004|loc=§13.4}}</ref> 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 <math>K=\mathbf{Q}(\sqrt{d})</math> (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 an algebraic number field|discriminant]] of ''K'', then the fundamental unit is
:<math>\varepsilon=\frac{a+b\sqrt{\Delta}}{2}</math>
where (''a'',&nbsp;''b'') is the smallest solution to<ref>{{harvnb|Neukirch|1999|loc=Exercise I.7.1}}</ref>
:<math>x^2-\Delta y^2=\pm4</math>
in positive integers. This equation is basically [[Pell's equation]] or the negative Pell equation and its solutions can be obtained similarly using the [[simple continued fraction|continued fraction expansion]] of <math>\sqrt{\Delta}</math>.

Whether or not ''x''<sup>2</sup>&nbsp;−&nbsp;Δ''y''<sup>2</sup>&nbsp;=&nbsp;−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 <math>\sqrt{\Delta}</math> 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''&nbsp;=&nbsp;34.<ref>{{harvnb|Alaca|Williams|2004|loc=Table 11.5.4}}</ref> 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''<sup>−</sup>(''X'') is those who have a unit of norm −1, then<ref>{{harvnb|Stevenhagen|1993|loc=Conjecture 1.4}}</ref>
:<math>\lim_{X\rightarrow\infty}\frac{D^-(X)}{D(X)}=1-\prod_{j\geq1\text{ odd}}\left(1-2^{-j}\right).</math>
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<ref>{{harvnb|Fouvry|Klüners|2010}}</ref> who show that the converse fails between 33% and 59% of the time.  In 2022, Peter Koymans and Carlo Pagano<ref>{{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 }}</ref> 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|&nbsp;>&nbsp;1 in this embedding. If the discriminant Δ of ''K'' satisfies |Δ|&nbsp;≥&nbsp;33, then<ref>{{harvnb|Alaca|Williams|2004|loc=Theorem 13.6.1}}</ref>
:<math>\epsilon^3>\frac{|\Delta|-27}{4}.</math>
For example, the fundamental unit of <math>\mathbf{Q}(\sqrt[3]{2})</math> is <math>\epsilon = 1+\sqrt[3]{2}+\sqrt[3]{2^2},</math> and <math>\epsilon^3\approx 56.9</math> whereas the discriminant of this field is −108 thus
:<math>\frac{|\Delta|-27}{4}=20.25</math>
so <math>\epsilon^3 \approx 56.9 > 20.25</math>.

==Notes==
{{reflist}}

==References==
*{{Citation
| last=Alaca
| 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]]