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Class number of (v, n, M)-extensions

Alkam, O
Bilhan, M
An analogue of cyclotomic number fields for function fields over the finite held F-q was investigated by L. Carlitz in 1935 and has been studied recently by D. Hayes, M. Rosen, S. Galovich and others. For each nonzero polynomial M in F-q[T], we denote by k(Lambda (M)) the cyclotomic function field associated with M, where k = F-q(T). Replacing T by 1/T in k and considering the cyclotomic function held F-upsilon that corresponds to (1/T)(upsilon +1) gets us an extension of k, denoted by L-upsilon, which is the fixed field of F-upsilon module F-q*. We define a (upsilon, n, M)-extension to be the composite N = k(n)k(Lambda (m))L-upsilon where k(n) is the constant field of degree n over k. In this paper we give analytic class number formulas for (upsilon, n, M)-extensions when M has a nonzero constant term.