On ramification in the compositum of function fields
Anbar, Nurdagül and Stichtenoth, Henning and Tutdere, Seher (2009) On ramification in the compositum of function fields. Bulletin of the Brazilian Mathematical Society, 40 (4). pp. 539-552. ISSN 1678-7544 (Print) 1678-7714 (Online)
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Official URL: http://dx.doi.org/10.1007/s00574-009-0026-8
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.
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