Real quadratic fields
For the real quadratic field (with ''d'' square-free), the fundamental unit ε is commonly normalized so that (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 : 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. 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 : 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 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 : For example, the fundamental unit of is and whereas the discriminant of this field is −108 thus : so .Notes
References
* * * * *External links
* {{MathWorld, title=Fundamental Unit, urlname=FundamentalUnit Algebraic number theory