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Explicit Form of Cassels’ p-adic Embedding Theorem for Number Fields

Published online by Cambridge University Press:  20 November 2018

Arturas Dubickas ,
Min Sha  and
Igor Shparlinski
Arturas Dubickas
Affiliation:
Department of Mathematics and Informatics, Vilnius University, LT-03225 Vilnius, Lithuania. e-mail: arturas.dubickas@mif.vu.lt
Min Sha
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia. e-mail: shamin2010@gmail.com, igor.shparlinski@unsw.edu.au
Igor Shparlinski
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia. e-mail: shamin2010@gmail.com, igor.shparlinski@unsw.edu.au
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Abstract

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In this paper, we give a general explicit form of Cassels’ $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.

Keywords

11R0411S8511G5011R0911R18number fieldp-adic embeddingheightpolynomialcyclotomic field
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 5 , 01 October 2015 , pp. 1046 - 1064
Copyright
Copyright © Canadian Mathematical Society 2015

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