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Mathematics of Computation

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Local computation of differents and discriminants

Author: Enric Nart
Journal: Math. Comp. 83 (2014), 1513-1534
MSC (2010): Primary 11Y40; Secondary 11Y05, 11R04, 11R27
Published electronically: July 31, 2013
MathSciNet review: 3167470
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Abstract: We obtain several results on the computation of different and discriminant ideals of finite extensions of local fields. As an application, we deduce routines to compute the $ \mathfrak{p}$-adic valuation of the discriminant $ \operatorname {Disc}(f)$, and the resultant $ \operatorname {Res}(f,g)$, for polynomials $ f(x),g(x)\in A[x]$, where $ A$ is a Dedekind domain and $ \mathfrak{p}$ is a non-zero prime ideal of $ A$ with finite residue field. These routines do not require the computation of either $ \operatorname {Disc} (f)$ or $ \operatorname {Res}(f,g)$; hence, they are useful in cases where this latter computation is inefficient because the polynomials have a large degree or very large coefficients.

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Additional Information

Enric Nart
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, E-08193 Bellaterra, Barcelona, Catalonia, Spain

Keywords: Different, discriminant, global field, local field, Montes algorithm, Newton polygon, Okutsu invariant, OM representation, resultant, Single-factor lifting algorithm
Received by editor(s): May 7, 2012
Received by editor(s) in revised form: September 5, 2012
Published electronically: July 31, 2013
Additional Notes: Partially supported by MTM2009-10359 from the Spanish MEC
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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