A complete study of the ramification for any separable cubic global function field (Q2336059)

The authors proved in [Eur. J. Math. 5, No. 2, 551--570 (2019; Zbl 1429.11193)] that any separable cubic extension \(L=K(y)\) of an arbitrary field \(K\) of characteristic \(p\geq 0\) is given by one of the following generating equations, each one depending on one parameter: (a) \(y^3=a\), with \(a\in K\) and \(p\neq 3\); (b) \(y^3-3y=a\), with \(a\in K\) and \(p\neq 3\); (c) \(y^3+ay+a^2=0\), with \(a\in K\) and \(p=3\). In the paper under review, they apply their result to a cubic extension \(L/K\) where \(K\) is a global function field and find: (1) when \(L/K\) is an extension of constants in Section 3; (2) the ramification of any place of \(K\) in \(L\) when \(L/K\) is a geometric extension in Section 4; and (3) the genus \(g_L\) in terms of \(g_K\) that is, the genus Riemann-Hurwitz formula in Section 5. In each case, (1), (2), and (3), the authors consider each type of generating equation (a), (b) and (c). With this paper, a complete study of the ramification of any separable cubic extension of \(L/K\) of global function fields is given. A similar study may be carried out using the corresponding Tate genus formula for inseparable extensions [Proc. Am. Math. Soc. 3, 400--406 (1952; Zbl 0047.03901)].