The euclidean algorithm for number fields and primitive roots

Let K be a number field with unit rank at least four, containing a subfield M such that K/M is Galois of degree at least four. We show that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann Hypothesis for Dedekind zeta functions. We prove this unconditionally.