Generators for certain cyclic extensions of global function fields (Q1801898)

Let \(k\) be a finite field with \(q=p^ r\) elements. The group of automorphisms of the rational function field \(k(x)/k\) is isomorphic to \(\text{PGL}(2,k)\). Let \(A\in \text{PGL}(2,k)\). If \(A\) is represented by the matrix \(M={{a\;b} \choose {c\;d}}\), then \(A\) induces an automorphism of \(k(x)\) which is specified by \(x^ A=x^ M= (ax+b)/(cx+d)\). In this paper, the author considers to what extent this situation is generic. Let \(L\) be an algebraic function field in one variable with exact field of constants \(k\). Let \(\sigma\) be an automorphism of \(L/k\) with finite order. Let \(K\) be the fixed field of \(\sigma\). The author asks the question: is \(L\) generated over \(K\) by an element \(x\) such that \(x^ \sigma\) is a linear functional transformation of \(x\)? Let \(n\) denote the order of \(\sigma\). The author deals with the case \(n\mid q+1\) with \(n>2\) since the answer is known in other cases. He proves the following results. Let \(n\) be an odd integer with \(n\mid q+1\) and \(n>2\). Let \(A\in \text{PGL}(2,k)\) of order \(n\). Let \(L/K\) be a cyclic geometric extension of degree \(n\). Let \(\sigma\) be a generator of \(\text{Gal}(L/K)\). Then \(L\) is generated over \(K\) by an element \(x\) with \(x^ \sigma= x^ A\). Let \(n\) be an even integer with \(n\mid q+1\) and \(n>2\). Let \(A\in \text{PGL}(2,k)\) of order \(n\). Let \(L/K\) be a cyclic geometric extension of degree \(n\). Let \(\sigma\) be a generator of \(\text{Gal}(L/K)\). Then \(L\) is generated over \(K\) by an element \(x\) with \(x^ \sigma=x^ A\) if and only if the primes of \(K\) ramified in \(L/K\) have even degrees.