2011/09/21 by Maurizio Monge, Monge, Maurizio · 1 citation
Computer Science · Mathematics · #11S15 #11S31 #11Y40 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1109.4617
openalex publication_date 2011/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a local field with finite residue field, we define a normal form\nfor Eisenstein polynomials depending on the choice of a uniformizer \πK and\nof residue representatives. The isomorphism classes of extensions generated by\nthe polynomials in the family exhaust all totally ramified extensions, and the\nmultiplicity with which each isomorphism class L/K appears is always smaller\nthan the number of conjugates of L over K.\n An algorithm to recover the set of all special polynomials generating the\nextension determined by a general Eisenstein polynomial is described. We also\ngive a criterion to quickly establish that a polynomial generates a different\nextension from that generated by a set of special polynomials, such criterion\ndoes not only depend on the usual distance on the set of Eisenstein polynomials\nconsidered by Krasner and others.\n We conclude with an algorithm for the construction of the unique special\nequation determining a totally ramified class field in general degree, given a\nsuitable representation of a group of norms.\n