Fundamental unit (number theory)

In algebraic number theory, a fundamental unit is a generator for the torsion-free unit group of the ring of integers of a number field, when that group is infinite cyclic. See also Dirichlet's unit theorem.

For rings of the form $\mathbb{Z}[\sqrt n]$, the fundamental unit has the form $x+y\sqrt n,$ 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. pp. 92–93. ISBN 0-387-97037-1.