Artin's Primitive Root Conjecture and the Euclidean Algorithm

Artin's famous primitive root conjecture states that if n is an integer
other than -1 or a square, then there are infinitely many primes p such
that n is a primitive root modulo p. I will discuss a number field version
of this conjecture and its connection to the following Euclidean algorithm
problem. Let O be the ring of integers of a number field K. It is
well-known that if O is a Euclidean domain, then O is a unique
factorization domain. With the exception of the imaginary quadratic number
fields, it is conjectured that the reverse implication is true. This was
proven with the assumption of the GRH by Weinberger; I will discuss some
recent work in proving the conjecture (unconditionally) for infinitely
many number fields. This is joint work with M. Ram Murty.