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. We 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 is joint work with M. Ram Murty.