Gau periods have been used successfully as a tool for constructing normal bases in nite elds. Starting from a primitive rth root of unity, one obtains under certain conditions a normal basis for F q n over F q , where r is a prime and nk = r ? 1 for some integer k. We generalize this construction by allowing arbitrary integers r with nk = '(r), and nd in many cases smaller values of k than is possible with the previously known approach. The rst two authors are with Fachbereich17