Higher newton polygons and integral bases

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Abstract

Let A be a Dedekind domain whose field of fractions K is a global field. Let p be a non-zero prime ideal of A, and Kp the completion of K at p. The Montes algorithm factorizes a monic irreducible polynomial f∈. A[. x] over Kp, and provides essential arithmetic information about the finite extensions of Kp determined by the different irreducible factors. In particular, it can be used to compute a p-integral basis of the extension of K determined by f. In this paper we present a new and faster method to compute p-integral bases, based on the use of the quotients of certain divisions with remainder of f that occur along the flow of the Montes algorithm.

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Guàrdia, J., Montes, J., & Nart, E. (2015). Higher newton polygons and integral bases. Journal of Number Theory, 147, 549–589. https://doi.org/10.1016/j.jnt.2014.07.027

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