On ramification in the compositum of function fields (Q1041288)

\textit{H. Hasse} [J. Reine Angew. Math. 172, 37--54 (1934; Zbl 0010.00501)] gave the explicit formulas for ramification and different exponents in cyclic extension of function fields. The authors propose the following generalization of Hasse's result. Let \(F/K\) be a function field with perfect constant field \(K\), and let \(E=F(y)\) be its extension, where \(y\) satisfies \(f(y) = u g(y)\) with \(f(y), g(y) \in K[y]\) and \(u \in F \setminus K\). Let \(P'\) be a place of \(E\). Set \(P := P'\cap F, Q := P' \cap K(u),Q' : = P' \cap K(y), e_0 := e(P|Q), e := e(Q'|Q), d := d(Q'|Q), r := \mathrm{gcd}(e_0,e)\). If not both \(P|Q\) and \(Q'|Q\) are wild, then (a) \(e(P'|P) = e/r\). In particular, if \(Q'|Q\) is tame then \(P'| P\) is also tame and hence the different exponent is \(d(P'|P) = e(P'|P) -1 = e/r-1\). (b) If \(P|Q\) is tame, then \(d(P'|P) = (e_0(d+1-e) +e)r^{-1} -1\). Hasse's formulas for ramification and different exponents in Kummer and Artin-Schreier extensions are special cases of the above result. Its proof is based on Abhyankar's Lemma on the ramification in the compositum \(E=E_1E_2\) of finite extensions of a function field \(F\). That lemma does not hold if both \(E_1/F\) and \(E_2/F\) are wildly ramified. Another main result of the paper is a generalization of Abhyankar's Lemma to the case where \(E_1/F\) and \(E_2/F\) are cyclic of degree \(p= \mathrm{char}\,(K)\). Certainly, it will be useful in the study of wild towers of function fields.