Vol. 13, No. 5, 2020

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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 and Scott Zinzer

Vol. 13 (2020), No. 5, 747–758

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(n 1), 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.

$p$-adic field, Eisenstein polynomial, discriminant, invariant
Mathematical Subject Classification
Primary: 11S05, 11S15
Received: 22 April 2019
Revised: 17 February 2020
Accepted: 15 September 2020
Published: 5 December 2020

Communicated by Kenneth S. Berenhaut
Chad Awtrey
Department of Mathematics and Statistics
Elon University
Elon, NC
United States
Alexander Gaura
Department of Mathematics
Princeton University
Princeton, NJ
United States
Sebastian Pauli
Department of Mathematics and Statistics
University of North Carolina
Greensboro, NC
United States
Sandi Rudzinski
Department of Mathematics and Statistics
University of North Carolina
Greensboro, NC
United States
Ariel Uy
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA
United States
Scott Zinzer
Department of Mathematics
Aurora University
Aurora, IL
United States