Primitive normal bases with prescribed trace

Let E be a finite degree extension over a finite field F = GF(q), G the Gal ois group of E over F and let a E F be nonzero. We prove the existence of a n element w in E satisfying the following conditions: w is primitive in E, i.e., w generates the multiplicative group of E (as a module over the ring of integers). the set {w(g) \g is an element of G} of conjugates of w under G forms a nor mal basis of E over F. the (E, F)-trace of w is equal to a. This result is a strengthening of the primitive normal basis theorem of Len stra and Schoof [10] and the theorem of Cohen on primitive elements with pr escribed trace [3]. It establishes a recent conjecture of Morgan and Mullen [14], who, by means of a computer search, have verified the existence of s uch elements for the cases in which q less than or equal to 97 and n less t han or equal to 6, n being the degree of E over F. Apart from two pairs (F, E) (or (q, n)) we are able to settle the conjecture purely theoretically.