Generic polynomials for cyclic function field extensions over certain finite fields (Q723351)

Let \(K\) be a congruence function field over the finite field of \(q\) elements. Let \(L/K\) be a cyclic extension of degree \(l\), where \(l\) is a positive integer such that \(q\equiv -1\bmod l\). Note that \(L/K\) is not a Kummer extension. Let \(\xi\) be a primitive \(l\)-th root of unity. Then \([{\mathbb F}_q(\xi):{\mathbb F}_q]=2\) and \(\xi+\xi^{-1}\in{\mathbb F}_q\). Let \({\mathrm{Gal}}({\mathbb F}_q(\xi)/{\mathbb F}_q)=\langle\sigma\rangle\) with \(\sigma\xi=\xi^{-1}\). The extension \(L(\xi)/F(\xi)\) is a Kummer extension. The aim of this paper is to find generic polynomials for geometric cyclic function field extensions over \({\mathbb F}_q\), including the classification of cyclic extensions up to isomorphism over \({\mathbb F}_q\). The main result, Theorem 1.5, says that there exists a Kummer generator \(\omega\) for \(L(\xi)/K(\xi)\), whose minimal polynomial is \(X^l-a\) such that \(y:=\sigma\omega+\omega\) is a generator for \(L/K\) and \(u:=\omega \sigma \omega\in K\) so that the minimal polynomial of \(y\) over \(K\) is \[P_{u,\alpha}^l(x)=x^l-lux^{l-2}+\sum_{s=1}^{[l/2]} c_{s,l} u^s x^{l-2s}-\alpha, \] where \(\alpha=a+\sigma a\), \(u^l=a\sigma a\) and some explicit coefficients \(c_{s,j}\) computed recursively. Conversely, if \(L/K\) is a geometric extension of degree \(l\) and a generator \(y\) whose minimal polynomial is of the form \(P_{u,\alpha}^l (x)\) where \(u,\alpha\in K\) are such that \(u^l=a\sigma a\) and \(\alpha= a+\sigma a\) for some \(a\in K(\xi)\setminus K\), then \(L/K\) is a cyclic extension. Next the author obtains a classification of geometric cyclic extensions of degree \(l\) with \(q\equiv -1\bmod l\) similar to that of Kummer extensions. Finally, the second main result is an explicit description of the ramification in \(L/K\) of any prime \({\mathfrak p}\) in \(K\), Theorem 2.1.