For rings of the form $mathbb\{Z\}[sqrt\; n]$, the fundamental unit has the form $x+ysqrt\; n,$ where (x,y) is the smallest nontrivial solution to Pell's equation x² - ny² = 1 or x² - ny² = -1. (For example, if n is a prime that is 1 mod 4 then the fundamental unit comes from a solution of the second equation.)

References

• Duncan Buell (1989). Binary quadratic forms: classical theory and modern computations. Springer-Verlag.