Class number in constant extensions of function fields

Abstract

Let F/K be a in one variable of genus g having the finite K as exact of constants. Suppose p is a rational prime not dividing the class number of F. In this paper an upper bound is derived for the degree of a constant extension E necessary to have p occur as a divisor of the class number of the E. Throughout this paper the term function field will mean a in one variable whose exact of constants is a finite with q elements. Let F/K be a field. The order of the finite group of divisor classes of degree zero is the class number hf. For F/K of genus g, we use the notation of [2] and denote by L(u) the polynomial numerator of the zeta of F. It follows from the functional equation of the zeta that (1) L(u)1? + a1u + a2u2 + * * + ag u + qag_lu9l + * + qg-lalu2g-l + q U2g and L(u) & Z[u], Z the rational integers. Furthermore the class number hBF=L(1). If E/F is a constant extension of degree n, then the polynomial numerator L,(u) of the zeta for E is given by (2) L,(u) = 1 + b1u + * * * + bg u9 + qnbg_1u9l + * * * + q nu2g where the coefficients by (j= 1, ... ,g) are, with appropriate sign, the elementary symmetric functions of the nth powers of the reciprocals of the roots of (1). The genus of E is the same as that of F because F is conservative. In this paper we give an upper bound for the degree of a constant extension E of F necessary to have a predetermined prime p occur as a Received by the editors September 10, 1971 and, in revised form, March 12, 1972. AMS 1969 subject classifications. Primary 1078; Secondary 1278, 1435.