Crossed product algebras defined by separable extensions

We generalize the classical construction of crossed product algebras defined by finite Galois field extensions to finite separable field extensions. By studying properties of rings graded by groupoids, we are able to calculate the Jacobson radical of these algebras. We use this to determine when the analogous construction of crossed product orders yield Azumaya, maximal, or hereditary orders in a local situation. Thereby we generalize results by Haile, Larson, and Sweedler.