On ramification in the compositum of function fields

Abstract

The aim of this paper is twofold: Firstly, we generalize well-known formulas for ramification and different exponents in cyclic extensions of function fields over a field K (due to H. Hasse) to extensions E = F(y), where y satisfies an equation of the form f(y) = u · g(y) with polynomials f(y), g(y) ∈ K[y] and u ∈ F. This result depends essentially on Abhyankar’s Lemma which gives information about ramification in a compositum E = E ₁ E ₂ 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 wildly ramified. Our second objective is a generalization of Abhyankar’s Lemma if 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.