Primitivity, freeness, norm and trace

Given the extension E/F of Galois fields, where F = GF(q) and E = GF(q^n), we prove that, for any primitive b element of F*, there exists a primitive element in E which is free over F and whose (E, F)-norm is equal to b. Furthermore, if (q,n) unequal (3,2), we prove that, for any nonzero b element of F, there exists an element in E which is free over F and whose (E,F)-norm is equal to b. A preliminary investigation of the question of determining whether, in searching for a primitive element in E that is free over F, both the (E,F)-norm and the (E,F)-trace can be prescribed is also made: this is so whenever n>=9