- 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 anumber field , when that group isinfinite cyclic . See alsoDirichlet's unit theorem .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 "x"^{2}- "ny"^{2}= 1 or "x"^{2}- "ny"^{2}= -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***

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