Calculating formula and divisibility for relative class numbers of abelian function fields (Q838457)

Let \(P\) be an irreducible polynomial of degree \(d>0\) over a finite field \(F_q\) of odd characteristic \(p\). Let \(L\) be the cyclotomic function field corresponding to \(P\). Let \(K\) be a subextension of \(L\) containing the maximal real subfield \(L^+\) of the field \(L\). The main result of the paper is an explicit determinant formula for the relative divisor class number \(h^{-}_K=h_K/h_{L^+}\) of the field \(K.\) Using this formula the authors give a criterion when the relative divisor class number is divisible by the characteristic \(p.\)