Galois (h-Galois) and locally fnute. However, we have believed that [7, Theorem 2.1] and [7, Theorem 3.1] should be stated under more desirable assumptions.

Abstract. As is well known, N?(«) = (1/n) J2d\n ß{d)q"ld coincides with the number of monic irreducible polynomials of GF(q)[X] of degree n . In this note we discuss the curve nNx(n) and the solutions of equations nNx(n) = b (b > n) . As a corollary of these results, we present a necessary and sufficient arithmetical condition for R/K to have a primitive element.

In [5], we can find some fundamental theorems of quasi-Galois extensions. The purpose of the present paper is to expose several additional theorems concerning such extensions. At first, we shall recall the following lemmas which have been obtained in [4] and [5].

Throughout the present paper, R will be a simple ring, where we shall understand by a simple ring a total matrix ring over a division rings. If S' is any subring containing the identity element 1 of R, we denote by

If a simple ring R is Galois and finite over a simple subring S then R-Su, v] with some conjugate u, v 6, Theorem 1]. In case R is a division ring we have seen that R--Sr with some r if and only if R is commutative or S is not contained in the center C of R [_2, Theorem 3. The purpose of this paper is to prove that this fact is still valid for simple rings.Throughout the present paper, R be always a simple ring (with minimum condition), and S a simple subring of R containing the identity element of R. And we shall use the following conventions" R--De, where e's are matrix units and D--V({e.'s}) a division ring. And, C, Z and V are the center of R, the center of S and the centralizer V(S)of S in R respectively. When R is Galois over S, we denote by ( the Galois group of R/S. Finally, as to notations and terminologies used in this paper, we follow 4.