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On ramification in the compositum of function fields

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 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

On ramification in the compositum of function fields

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References (9)

Publisher
Springer Journals
Copyright
Copyright © 2009 by Springer
Subject
Mathematics; Theoretical, Mathematical and Computational Physics; Mathematics, general
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-009-0026-8
Publisher site
See Article on Publisher Site

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.

Journal

Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals

Published: Nov 10, 2009

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