Primitive roots and the Euclidean algorithm

An integer s is called a primitive root modulo a prime p if the multiplicative set generated by s surjects onto all non-zero residue classes modulo p. Artin's primitive root conjecture states that all integers s other than -1 or squares are primitive roots modulo infinitely many primes. I'll discuss a generalization of Artin's primitive root conjecture to number fields and connections this has to the Euclidean Algorithm problem. This is joint work with R. Murty.