08.01.2009

Using results of D.V.Chudnowsky, G.V.Chudnowsky [3] and W.C.Waterhouse [10] we prove that the rank (=bilinear complexity of multiplication) of the finite field F_qⁿ viewed as an F_q-algebra is 2n if n satisfies q/2 + 1 < n < (q + 1 + ε(q))/2. Here ε(q) is the greatest integer ≤ 2q^1/2 which is prime to q if q is not a perfect square and ε(q) = 2q^1/2 if q is a perfect square.