Determining irreducibility and ramification groups for an additive extension of the rational function field Original Research Article

Let K be a subfield of image, not necessarily proper, and a(T) be an additive polynomial defined over K. Suppose that f(x)set membership, variantK[x] and consider the polynomial a(T)−f(x) over K(x). We provide a method to (i) determine its irreducibility and (ii) compute the ramification groups in the resulting additive extension of K(x). In several cases, our method is easier to use and provides more information than the Artin–Schreier theorem. As an application of this method, we study families of additive extensions of K(x) obtained using symmetric polynomials. We prove irreducibility and determine their ramification groups, genera, and number of rational places. We show that these families contain examples of function fields with many rational places.