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.

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We present ''canonical forms'' of finite dimensional (quasi-Frobenius) commutative algebras L over a field k such that the radical cubed is zero and L modulo the radical is a product of copies of k. We also determine the isomorphism classes of the algebras L over some typical fields. Let X, Y 2 M n (k). Then X is said to be congruent to Y if there exists a P 2 GL n (k) such that X ¼ PY t P, where t P is the transpose of P.

We study to classify, up to isomorphism, algebras over a field k such that the radical cubed is zero and modulo the radical is a product of copies of k. The number of local quasi-Frobenius k-algebras with the condition is shown to be not less than the cardinality of k. In particular, the canonical forms of those algebras of dimension 5 are presented and their isomorphism classes are completely determined under some conditions on k.

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