Primitive complete normal bases for regular extensions

The extension E of degree n over the Galois field F = GF(q) is called regular over F, if ord_r(q) and n have greatest common divisor 1 for all prime divisors r of n which are different from the characteristic p of F (here, ord_r(q) denotes the multiplicative order of q modulo r). Under the assumption that E is regular over F and that q - 1 is divisible by 4 if q is odd and n is even, we prove the existence of a primitive element w element of E which is also completely normal over F (the latter means that w simultaneously generates a normal basis for E over every intermediate field K of E/F). Our result achieves, for the class of extensions under consideration, a common generalization of the theorem of Lenstra and Schoof on the existence of primitive normal bases [12] and the theorem of Blessenohl and Johnsen on the existence of complete normal bases [1]