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2020 The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of $p$-adic fields
Chad Awtrey, Alexander Gaura, Sebastian Pauli, Sandi Rudzinski, Ariel Uy, Scott Zinzer
Involve 13(5): 747-758 (2020). DOI: 10.2140/involve.2020.13.747

Abstract

Let K be an extension of the p-adic numbers with uniformizer π. Let φ and ψ be Eisenstein polynomials over K of degree n that generate isomorphic extensions. We show that if the cardinality of the residue class field of K divides n(n1), then v(disc(φ) disc(ψ))>v(disc(φ)). This makes the first (nonzero) digit of the π-adic expansion of disc(φ) an invariant of the extension generated by φ. Furthermore we find that noncyclic extensions of degree p of the field of p-adic numbers are uniquely determined by this invariant.

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Chad Awtrey. Alexander Gaura. Sebastian Pauli. Sandi Rudzinski. Ariel Uy. Scott Zinzer. "The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of $p$-adic fields." Involve 13 (5) 747 - 758, 2020. https://doi.org/10.2140/involve.2020.13.747

Information

Received: 22 April 2019; Revised: 17 February 2020; Accepted: 15 September 2020; Published: 2020
First available in Project Euclid: 22 December 2020

MathSciNet: MR4190435
Digital Object Identifier: 10.2140/involve.2020.13.747

Subjects:
Primary: 11S05, 11S15

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.13 • No. 5 • 2020
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