Primitive roots in a finite field

Abstract

with minimum m such that a(Qy) =0; y is said to belong to a(x). In particular if a(x) =Xpnx, Ore calls y a primitive root. It is easy to see that the two varieties of primitive roots are not identical; for example in the GF(24) defind by 04+0+1, 0 is a primitive root in the original sense but not in Ore's sense (since it belongs to x8+x4+x2+x). On the other hand it can be verified that 03 is primitive according to Ore but belongs to the numerical exponent 5. To avoid confusion we shall refer to ordinary primitive roots as primitive roots of the first kind, while those satisfying Ore's definition will be called roots of the second kind.-Ore proved that primitive roots of the second kind exist; indeed there are precisely b(Xn-1)CGF(pn), where 4 now denotes the Euler function for GF[p, x]. The equivalent result in terms of the existence of a normal basis (see ?2) had been proved by Hensel. It is natural to ask whether one can find a number 3EGF(pn) which is simultaneously a primitive root of both the first and second kinds. More generally if eI pn 1 and a(x) I Xpn_x, can one find a number ,B belonging to the numerical exponent e and the linear polynomial a(x)? We shall show that the first question is answered in the affirmative for pn sufficiently large; the second question also admits of an affirmative answer provided pn is large and e deg a(x) is sufficiently large. The method of proof is suggested by the proof of Vinogradoff's theorem that the least primitive root of a prime p is Q(pl/2+E); see [5, p. 178], also [3]. In the opposite direction we show (Theorem 4) that for given p, r there exist infinitely many irreducible polynomials P such that no polynomial of degree < r can be a primitive root of the second kind (mod P). Finally (Theorem 6) we obtain a bound for