pith. sign in

arxiv: 2606.01453 · v1 · pith:GHMPUGUVnew · submitted 2026-05-31 · 🧮 math.NT

Distinguished defining polynomials for extensions of p-adic fields

Jordi Gu\`ardia i R\'ubies , John W. Jones , Kevin Keating , Sebastian Pauli , David P. Roberts , David Roe This is my paper

Pith reviewed 2026-06-28 16:11 UTC · model grok-4.3

classification 🧮 math.NT
keywords p-adic fieldsdefining polynomialsfield extensionsalgorithmslocal number fieldsdatabases
0
0 comments X

The pith

An algorithm produces a distinguished defining polynomial for any p-adic field extension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies an explicit algorithm that selects one special minimal polynomial to define a given extension of the p-adic numbers. The selection rule is deterministic, so the same field extension receives the same polynomial no matter who runs the procedure. This standardization was used to enlarge a public database of p-adic fields. A reader would care because it removes the need for ad-hoc choices when labeling or comparing these local number fields in computations.

Core claim

We give an algorithm for choosing a distinguished defining polynomial for a p-adic field extension. This algorithm formed an important ingredient in the recent expansion of the database of p-adic fields within the L-functions and modular forms database.

What carries the argument

The algorithm that selects the distinguished defining polynomial by applying a deterministic sequence of tests to possible minimal polynomials.

If this is right

  • Every p-adic extension receives a unique label that can be reproduced by anyone.
  • Database records for the same field become identical across independent computations.
  • Comparisons and searches over collections of p-adic fields become unambiguous.
  • Further algorithmic work on p-adic fields can assume a fixed defining polynomial without loss of generality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same selection rule could be applied to standardize presentations of extensions over other local fields, such as function fields over finite fields.
  • If the algorithm is fast enough, it could be embedded in computer-algebra systems to auto-generate canonical models on demand.
  • Consistent labeling may simplify the statement and verification of conjectures that involve counting or averaging over p-adic fields.

Load-bearing premise

That a single defining polynomial can be singled out by a rule that depends only on the field and not on any external ordering or arbitrary tie-breaker.

What would settle it

Running the algorithm on a concrete extension and obtaining two different output polynomials.

Figures

Figures reproduced from arXiv: 2606.01453 by David P. Roberts, David Roe, John W. Jones, Jordi Gu\`ardia i R\'ubies, Kevin Keating, Sebastian Pauli.

Figure 4.1
Figure 4.1. Figure 4.1: General form of the ramification polygon of a ν￾Øystein polynomial φ = ν(x) e + Pe−1 i=0 φ ∗ i (x)ν(x) i ∈ OK[x] of degree n = f · e = f · ϵpw. The segment of slope −1 only ex￾ists when f > 1 and a segment of slope 0 indicates ϵ > 1. f = 1, e = 18 13 8 1 2 5 0 1 7 2 −18 −9 −3 −1 0 13 6 3 f = 2, e = 9 7 −1 8 1 2 2 −18 −9 −3 −1 0 9 7 3 f = 3, e = 6 −1 0 5 2 −18 −6 −3 −1 0 12 5 f = 6, e = 3 −1 3 2 −18 −3 −1… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Ramification polygons of extensions L of degree 18 over Q3 with v3(disc(L/Q3)) = 30, also see the list of these families in the LMFDB [PITH_FULL_IMAGE:figures/full_fig_p007_4_2.png] view at source ↗
read the original abstract

We give an algorithm for choosing a distinguished defining polynomial for a p-adic field extension. This algorithm formed an important ingredient in the recent expansion of the database of p-adic fields within the L-functions and modular forms database.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts the existence of an algorithm for selecting a distinguished defining polynomial for a p-adic field extension and states that this algorithm was used as an ingredient in the recent expansion of the p-adic fields database in the L-functions and modular forms database (LMFDB).

Significance. A correctly specified, choice-independent algorithm for canonical polynomial selection would aid reproducibility and standardization in computational number theory databases. External corroboration via LMFDB deployment indicates practical utility if the method is made explicit and verifiable.

major comments (1)
  1. [Abstract] Abstract: the central claim is the provision of an algorithm, yet the manuscript supplies neither a description of the algorithm, a proof of its correctness or termination, nor any verification or example of its output. This renders the claim impossible to evaluate for soundness or independence from arbitrary choices.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim is the provision of an algorithm, yet the manuscript supplies neither a description of the algorithm, a proof of its correctness or termination, nor any verification or example of its output. This renders the claim impossible to evaluate for soundness or independence from arbitrary choices.

    Authors: We agree with the referee that the present manuscript does not contain a description of the algorithm, a proof of correctness or termination, or examples of its output. In the revised version we will supply a complete, self-contained description of the algorithm together with a proof that it terminates and produces a well-defined output independent of arbitrary choices, as well as explicit examples and verification that the output matches the polynomials used in the LMFDB expansion. revision: yes

Circularity Check

0 steps flagged

No significant circularity in algorithm presentation

full rationale

The paper describes an algorithm for selecting a distinguished defining polynomial for p-adic field extensions, used in LMFDB database expansion. This is a constructive, algorithmic contribution with no derivation chain, fitted parameters, or self-citation load-bearing steps visible in the provided text. The central claim is externally corroborated by database deployment rather than reducing to its own inputs by construction. No equations or uniqueness theorems are invoked in a self-referential manner.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not mention any free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5563 in / 1083 out tokens · 44098 ms · 2026-06-28T16:11:39.623847+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

127 extracted references · 55 canonical work pages

  1. [1]

    Alberich-Carramiñana, Maria and Gu\`ardia, Jordi and Nart, Enric and Poteaux, Adrien and Ro\'e, Joaquim and Weimann, Martin , TITLE =. Found. Comput. Math. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s10208-024-09646-x , URL =

    work page doi:10.1007/s10208-024-09646-x 2025

  2. [2]

    Eisenstein equations of degree

    Amano, Shigeru , Journal =. Eisenstein equations of degree. 1971 , Pages =

    1971

  3. [3]

    and Pollack, D

    Ash, A. and Pollack, D. and Sinnott, W. , Journal =. 2005 , Number =. doi:10.1016/j.jnt.2004.12.007 , Fjournal =

    work page doi:10.1016/j.jnt.2004.12.007 2005

  4. [4]

    2004 , Number =

    Ash, Avner and Pollack, David and Soares, Dayna , Journal =. 2004 , Number =

    2004

  5. [5]

    Involve , ISSN =

    Awtrey, Chad and Gaura, Alexander and Pauli, Sebastian and Rudzinski, Sandi and Uy, Ariel and Zinzer, Scott , Title =. Involve , ISSN =. 2020 , Language =. doi:10.2140/involve.2020.13.747 , Keywords =

    work page doi:10.2140/involve.2020.13.747 2020

  6. [6]

    Dodecic 3-adic fields , Author =. Int. J. Number Theory , Year =. doi:10.1142/S1793042112500558 , Fjournal =

    work page doi:10.1142/s1793042112500558

  7. [7]

    and Shill, Christopher and Strosnider, Erin , TITLE =

    Awtrey, Chad and Barkley, Brett and Miles, Nicole E. and Shill, Christopher and Strosnider, Erin , TITLE =. Rocky Mountain J. Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1216/RMJ-2015-45-6-1755 , URL =

    work page doi:10.1216/rmj-2015-45-6-1755 2015

  8. [8]

    Galois groups of degree 12 2-adic fields with trivial automorphism group , Author =

  9. [9]

    Involve , FJOURNAL =

    Awtrey, Chad and Miles, Nicole and Milstead, Jonathan and Shill, Christopher and Strosnider, Erin , TITLE =. Involve , FJOURNAL =. 2015 , NUMBER =. doi:10.2140/involve.2015.8.329 , URL =

    work page doi:10.2140/involve.2015.8.329 2015

  10. [10]

    Topics from the 9th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics , Year =

    Absolute resolvents and masses of irreducible quintic polynomials , Author =. Topics from the 9th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics , Year =

  11. [11]

    Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics , Year =

    Galois Groups of 2-Adic Fields of Degree 12 with Automorphism Group of Order 6 and 12 , Author =. Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics , Year =

  12. [12]

    Topics from the 9th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics , Year =

    A linear resolvent for degree 14 polynomials , Author =. Topics from the 9th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics , Year =

  13. [13]

    , Journal =

    Bauch, Jens-Dietrich and Nart, Enric and Stainsby, Hayden D. , Journal =. Complexity of. 2013 , Pages =. doi:10.1112/S1461157013000089 , Fjournal =

    work page doi:10.1112/s1461157013000089 2013

  14. [14]

    Zur allgemeinen

    Bauer, Mih\'aly , Journal =. Zur allgemeinen. 1907 , Pages =

    1907

  15. [15]

    , Author =

    Algorithmic proof of the epsilon constant conjecture. , Author =. Math. Comput. , Year =. doi:10.1090/S0025-5718-2013-02691-9 , Fjournal =

    work page doi:10.1090/s0025-5718-2013-02691-9 2013

  16. [16]

    Bosma, Wieb and Cannon, John and Playoust, Catherine , Journal =. The. 1997 , Note =. doi:10.1006/jsco.1996.0125 , Fjournal =

    work page doi:10.1006/jsco.1996.0125 1997

  17. [17]

    Beweis eines

    Brauer, Richard and Hasse, Helmut and Noether, Emmy , Journal =. Beweis eines. 1932 , Pages =

    1932

  18. [18]

    The transitive groups of degree up to eleven , Author =. Comm. Algebra , Year =. doi:10.1080/00927878308822884 , Fjournal =

    work page doi:10.1080/00927878308822884

  19. [19]

    Experiment

    The transitive permutation groups of degree 32 , Author =. Experiment. Math. , Year =

  20. [20]

    and Gordon, Daniel M

    Cantor, David G. and Gordon, Daniel M. , Booktitle =. Factoring polynomials over. 2000 , Pages =. doi:10.1007/10722028_10 , Mrclass =

    work page doi:10.1007/10722028_10 2000

  21. [21]

    La th\'orie du symbole de restes normiques , Author =. J. Reine Angew. Math. , Year =

  22. [22]

    Sur la th\'eorie du corps des classes dans les corps finis et les corps locaux , Author =. J. Fac. Sci. Univ. Tokyo Sect. I , Year =

  23. [23]

    1993 , PAGES =

    Cohen, Henri , TITLE =. 1993 , PAGES =. doi:10.1007/978-3-662-02945-9 , URL =

    work page doi:10.1007/978-3-662-02945-9 1993

  24. [24]

    and Hulpke, Alexander and McKay, John , TITLE =

    Conway, John H. and Hulpke, Alexander and McKay, John , TITLE =. LMS J. Comput. Math. , FJOURNAL =. 1998 , PAGES =

    1998

  25. [25]

    2018 , eprint=

    On enumerating extensions of p-adic fields with given invariants , author=. 2018 , eprint=

    2018

  26. [26]

    2005 , issn =

    Constructing transitive permutation groups , journal =. 2005 , issn =. doi:https://doi.org/10.1016/j.jsc.2004.08.002 , author =

    work page doi:10.1016/j.jsc.2004.08.002 2005

  27. [27]

    On transitive permutation groups , Author =. LMS J. Comput. Math. , Year =. doi:10.1112/S1461157000000115 , Fjournal =

    work page doi:10.1112/s1461157000000115

  28. [28]

    1993 , Series =

    Local Fields and Their Extensions , Author =. 1993 , Series =

    1993

  29. [29]

    2002 , Edition =

    Local Fields and Their Extensions , Author =. 2002 , Edition =

    2002

  30. [30]

    Computation of

    Fieker, Claus and Kl. Computation of. LMS J. Comput. Math. , Year =. doi:10.1112/S1461157013000302 , Fjournal =

    work page doi:10.1112/s1461157013000302

  31. [31]

    Implementing the round four maximal order algorithm , Author =. J. Th\'eor. Nombres Bordeaux , Year =

  32. [32]

    A fast algorithm for polynomial factorization over

    Ford, David and Pauli, Sebastian and Roblot, Xavier-Fran. A fast algorithm for polynomial factorization over. J. Th\'eor. Nombres Bordeaux , Year =

  33. [33]

    , Journal =

    Ford, David J. , Journal =. The construction of maximal orders over a. 1987 , Number =. doi:10.1016/S0747-7171(87)80054-8 , Fjournal =

    work page doi:10.1016/s0747-7171(87)80054-8 1987

  34. [34]

    and Veres, Olga , Booktitle =

    Ford, David J. and Veres, Olga , Booktitle =. On the complexity of the. 2010 , Pages =. doi:10.1007/978-3-642-14518-6_16 , Mrclass =

    work page doi:10.1007/978-3-642-14518-6_16 2010

  35. [35]

    Freundt, Sebastian and Karve, Aneesh and Krahmann, Anita and Pauli, Sebastian , Booktitle =. K. 2006 , Address =. doi:10.1007/11832225_15 , Mrclass =

    work page doi:10.1007/11832225_15 2006

  36. [36]

    Greve, Christian , Note =

  37. [37]

    Ramification polygons, splitting fields, and

    Greve, Christian and Pauli, Sebastian , Journal =. Ramification polygons, splitting fields, and. 2012 , Number =. doi:10.1142/S1793042112500832 , Fjournal =

    work page doi:10.1142/s1793042112500832 2012

  38. [38]

    Jones and Kevin Keating and Sebastian Pauli and David P

    Jordi Gu\`ardia and John W. Jones and Kevin Keating and Sebastian Pauli and David P. Roberts and David Roe. Families of p-adic Fields. LuCaNT: Databases, Algorithms, and Computational Number Theory. 2026

    2026

  39. [39]

    Arithmetic in big number fields: the '+Ideals' package , Author =

  40. [40]

    A new computational approach to ideal theory in number fields , Author =. Found. Comput. Math. , Year =. doi:10.1007/s10208-012-9137-5 , Fjournal =

    work page doi:10.1007/s10208-012-9137-5

  41. [41]

    Newton polygons of higher order in algebraic number theory , Author =. Trans. Amer. Math. Soc. , Year =. doi:10.1090/S0002-9947-2011-05442-5 , Fjournal =

    work page doi:10.1090/s0002-9947-2011-05442-5 2011

  42. [42]

    Gu. Higher. J. Th\'eor. Nombres Bordeaux , Year =. doi:10.5802/jtnb.782 , Fjournal =

    work page doi:10.5802/jtnb.782

  43. [43]

    Acta Arithmetica , Year =

    Okutsu Invariants and Newton Polygons , Author =. Acta Arithmetica , Year =

  44. [44]

    Single-factor lifting and factorization of polynomials over local fields , Author =. J. Symbolic Comput. , Year =. doi:10.1016/j.jsc.2012.03.001 , Fjournal =

    work page doi:10.1016/j.jsc.2012.03.001 2012

  45. [45]

    ArXiv e-prints , Year =

    Arithmetic in big number fields: the '+Ideals' package , Author =. ArXiv e-prints , Year =. 1005.4596 , Keywords =

    Pith/arXiv arXiv

  46. [46]

    2002 , Edition =

    Number theory , Author =. 2002 , Edition =

    2002

  47. [47]

    Hasse, Helmut , Journal =. Die. 1933 , Pages =

    1933

  48. [48]

    Hasse, Helmut , Journal =. Die. 1930 , Pages =

    1930

  49. [49]

    De nouveaux invariants num\'eriques pour les extensions totalement ramifi\'ees de corps locaux , Author =. J. Number Theory , Year =. doi:10.1006/jnth.1996.0092 , Fjournal =

    work page doi:10.1006/jnth.1996.0092 1996

  50. [50]

    , TITLE =

    Deligne, P. , TITLE =. Representations of reductive groups over a local field , SERIES =. 1984 , MRCLASS =

    1984

  51. [51]

    Helou, Charles , Publisher =. Non-. 1990 , Series =

    1990

  52. [52]

    Theorie der

    Hensel, Kurt , Publisher =. Theorie der. 1908 , City =

    1908

  53. [53]

    Ueber die

    Hensel, Kurt , Journal =. Ueber die. 1901 , Number =. doi:10.1007/BF01444976 , Language =

    work page doi:10.1007/bf01444976 1901

  54. [54]

    Hensel, Kurt , Journal =. \". 1897 , Pages =

  55. [55]

    Computing the multiplicative group of residue class rings , Author =. Math. Comp. , Year =. doi:10.1090/S0025-5718-03-01474-1 , Fjournal =

    work page doi:10.1090/s0025-5718-03-01474-1

  56. [56]

    1986 , Note =

    Local class field theory , Author =. 1986 , Note =

    1986

  57. [57]

    Iwasawa, Kenkichi , Journal =. On. 1955 , Pages =

    1955

  58. [58]

    , Journal =

    Jones, John W. , Journal =. Wild ramification bounds and simple group. 2011 , Number =. doi:10.1090/S0002-9939-2010-10628-7 , Fjournal =

    work page doi:10.1090/s0002-9939-2010-10628-7 2011

  59. [59]

    Number fields unramified away from 2 , Author =. J. Number Theory , Year =. doi:10.1016/j.jnt.2010.02.005 , Fjournal =

    work page doi:10.1016/j.jnt.2010.02.005 2010

  60. [60]

    Octic 2-adic fields , Author =. J. Number Theory , Year =. doi:10.1016/j.jnt.2007.12.014 , Fjournal =

    work page doi:10.1016/j.jnt.2007.12.014 2007

  61. [61]

    Galois number fields with small root discriminant , Author =. J. Number Theory , Year =. doi:10.1016/j.jnt.2006.05.001 , Fjournal =

    work page doi:10.1016/j.jnt.2006.05.001 2006

  62. [62]

    A database of local fields , Author =. J. Symbolic Comput. , Year =. doi:10.1016/j.jsc.2005.09.003 , Fjournal =

    work page doi:10.1016/j.jsc.2005.09.003 2005

  63. [63]

    Algorithmic number theory , Publisher =

    Nonic 3-adic fields , Author =. Algorithmic number theory , Publisher =. 2004 , Pages =. doi:10.1007/978-3-540-24847-7_22 , Mrclass =

    work page doi:10.1007/978-3-540-24847-7_22 2004

  64. [64]

    and Eick, Bettina and O'Brien, Eamonn A

    Holt, Derek F. and Eick, Bettina and O'Brien, Eamonn A. , TITLE =. 2005 , PAGES =. doi:10.1201/9781420035216 , URL =

    work page doi:10.1201/9781420035216 2005

  65. [65]

    Journal f\"ur die reine und angewandte

    K. Journal f\"ur die reine und angewandte. 1913 , Pages =

    1913

  66. [66]

    Proceedings of the 5th

    K. Proceedings of the 5th. 1912 , Pages =

    1912

  67. [67]

    Koch, Helmut , Journal =. \". 1965 , Pages =

    1965

  68. [68]

    1970 , Language =

    Koch, Helmut , Title =. 1970 , Language =

    1970

  69. [69]

    Remarques au sujet d'une note de

    Krasner, Marc , Journal =. Remarques au sujet d'une note de. 1979 , Number =

    1979

  70. [70]

    Nombre des extensions d'un degr\'e donn\'e d'un corps

    Krasner, Marc , Booktitle =. Nombre des extensions d'un degr\'e donn\'e d'un corps. 1966 , Pages =

    1966

  71. [71]

    Mathematica (Cluj) , Year =

    Sur la primitivit\'e des corps p-adiques , Author =. Mathematica (Cluj) , Year =

  72. [72]

    Standard generators of finite fields and their cyclic subgroups , JOURNAL =

    L. Standard generators of finite fields and their cyclic subgroups , JOURNAL =. 2023 , PAGES =. doi:10.1016/j.jsc.2022.11.001 , NOTE =

    work page doi:10.1016/j.jsc.2022.11.001 2023

  73. [73]

    Transactions of the American Mathematical Society , Year =

    p-adic power series which commute under composition , Author =. Transactions of the American Mathematical Society , Year =

  74. [74]

    , Journal =

    Lubin, Jonathan D. , Journal =. The local. 1981 , Number =

    1981

  75. [75]

    Transactions of the American Mathematical Society , Year =

    A construction for absolute values in polynomial rings , Author =. Transactions of the American Mathematical Society , Year =

  76. [76]

    Duke Mathematcial Journal , Year =

    A construction for prime ideals as absolute values of an algebraic field , Author =. Duke Mathematcial Journal , Year =

  77. [77]

    Miki, Hiroo , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1981 , PAGES =. doi:10.1515/crll.1981.328.99 , URL =

    work page doi:10.1515/crll.1981.328.99 1981

  78. [78]

    Topics from the 9th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics , Year =

    Constructing Splitting Fields of Polynomials over Local Fields , Author =. Topics from the 9th Annual UNCG Regional Mathematics and Statistics Conference, Springer Proceedings in Mathematics & Statistics , Year =

  79. [79]

    Computing

    Milstead, Jonathan , School =. Computing. 2017 , Url =

    2017

  80. [80]

    A family of

    Monge, Maurizio , Journal =. A family of. 2014 , Number =. doi:10.1142/S1793042114500511 , Fjournal =

    work page doi:10.1142/s1793042114500511 2014

Showing first 80 references.