Fundamental unit (number theory)

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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.)

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