In algebraic number theory

Algebraic number theory

Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

, a fundamental unit is a generator for the torsion-free unit group of the ring of integers

Algebraic integer

In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...

 of a number field, when that group is infinite cyclic. See also Dirichlet's unit theorem

Dirichlet's unit theorem

In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K...

.

For rings of the form , the fundamental unit has the form where (x,y) is the smallest nontrivial solution to Pell's equation

Pell's equation

Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this 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.)