Let Λ = (S/R,α) be a local weak crossed product order in the crossed product algebra A = (L/K,α) with integral cocycle, and $H=\sigma \in Gal(L/K)|\alpha (\sigma ,{\sigma }^{-1})\in S*$ the inertial group of α, for S* the group of units of S. We give a condition for the first ramification group of L/K to be a subgroup of H. Moreover we describe the Jacobson radical of Λ without restriction on the ramification of L/K.

We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish a complete list of norm-Euclidean such fields over imaginary quadratic number fields. This enables us to exhibit an example which gives a negative answer to a question asked by Eichler. The proofs are both theoretical and algorithmic.