On ramification in the compositum of function fields

The aim of this paper is twofold: Firstly, we generalize well-known formulas for ramification and different exponents in cycle extensions of function fields over a field K (due to H. Hasse) to extensions E = F (y), where y satisfies an equation of f (y) = u . g (y) with polynomials f (y), g (y) is an element of K [y] and u is an element of F. This result depends essentially on Abhyankar's Lemma which gives information about ramification in a compositum E = E1E2 of finite extensions E-1, E-2 over a function field F. Abhyankar's Lemma does not hold if both extensions E-1/F and E-2/F are widly ramified. Our second objective is a generalization of Abhyankar's Lemma E-1/F and E-2/F are cyclic extensions of degree p = char (K). This result may be useful for the study of wild towers of function fields over finite fields.