# Fundamental unit (number theory)

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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 $mathbb\left\{Z\right\} \left[sqrt n\right]$, 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.)

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