Hopf--Galois structures of cyclic type on parallel extensions of prime power degree

Andrew Darlington; Cindy Tsang

arxiv: 2510.14473 · v2 · submitted 2025-10-16 · 🧮 math.GR · math.NT· math.RA

Hopf--Galois structures of cyclic type on parallel extensions of prime power degree

Andrew Darlington , Cindy Tsang This is my paper

Pith reviewed 2026-05-18 06:36 UTC · model grok-4.3

classification 🧮 math.GR math.NTmath.RA

keywords Hopf-Galois structuresparallel extensionsprime power degreecyclic groupsGalois theorygroup theoretic methodsnormal closure

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The pith

If a prime power degree extension admits a cyclic Hopf-Galois structure, then all its parallel extensions do as well.

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

The paper asks whether the existence of a Hopf-Galois structure of type N on a finite separable extension L over K implies that every parallel extension sharing the same normal closure also has such a structure of type N. It provides a complete solution in the special case where the degree is a prime power and N is cyclic. The authors reduce the question to a group-theoretic problem using the Greither-Pareigis and Byott correspondences, which link Hopf-Galois structures to subgroups of the Galois group of the normal closure. A reader might care because this determines if the Hopf-Galois property is a feature of the normal closure rather than the specific choice of intermediate field.

Core claim

For any finite separable extension L/K of prime power degree that admits a Hopf-Galois structure of cyclic type N, every parallel extension L'/K also admits a Hopf-Galois structure of type N. This is shown by establishing that the group-theoretic conditions for the existence of the relevant subgroups or homomorphisms in the Galois group are satisfied simultaneously for all such parallel fields when the degree is a prime power and N is cyclic.

What carries the argument

The Greither-Pareigis and Byott correspondence, which translates the existence of a Hopf-Galois structure of type N on L/K into the existence of a certain subgroup of the Galois group of the normal closure or a homomorphism with prescribed properties.

If this is right

If one parallel extension admits the structure, all others of the same degree in the normal closure do too.
The property is determined by the Galois group of the normal closure rather than the specific subfield chosen.
Classification of cyclic Hopf-Galois structures on prime power extensions can focus on a single representative without loss of generality.
The result holds uniformly across all such extensions sharing the normal closure.

Where Pith is reading between the lines

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

This invariance might suggest similar uniformity for other types N if analogous group conditions can be verified.
Computing Hopf-Galois structures for families of extensions could be reduced by checking only one member of each parallel class.
Connections to other questions in Galois module theory or Hopf algebra actions on fields may benefit from this uniformity.

Load-bearing premise

The Greither-Pareigis and Byott translation of the Hopf-Galois question into group theory about the Galois group applies in the same way to every parallel extension at once.

What would settle it

Finding a specific example of a prime power degree separable extension with normal closure where one intermediate field of that degree has a cyclic Hopf-Galois structure but another does not.

read the original abstract

Let $L/K$ be any finite separable extension with normal closure $\widetilde{L}/K$. An extension $L'/K$ is said to be $\textit{parallel to $L/K$}$ if $L'$ is an intermediate field of $\widetilde{L}/K$ with $[L':K]=[L:K]$. We study the following question -- Given that $L/K$ admits a Hopf--Galois structure of type $N$, does it imply that every extension parallel to $L/K$ also admits a Hopf--Galois structure of type $N$? We completely solve this problem when the degree $[L:K]$ is a prime power and the type $N$ is cyclic. Our approach is group-theoretic and uses the work of Greither--Pareigis and Byott.

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.

Desk Editor's Note private letter to a colleague

Darlington and Tsang settle the parallel-extensions question for cyclic Hopf-Galois structures when the degree is a prime power by reducing it to a uniform subgroup existence condition in the Galois group.

read the letter

The main thing to know is that this paper gives a complete answer for the cyclic case when the degree is a prime power: if one extension admits a Hopf-Galois structure of cyclic type, then every parallel extension of the same degree does too. They work entirely in the Galois group of the normal closure and use the standard Greither-Pareigis and Byott correspondences to turn the field question into one about finding a cyclic subgroup T of order p^k that complements any given core-free H of index p^k with TH = G and T ∩ H trivial. The new contribution is showing this holds uniformly across all such H, not just for a single extension or conjugate class. That fills a specific gap left by earlier classification work on Hopf-Galois structures. The approach stays close to established tools and does not introduce new machinery, which keeps the argument clean and checkable. The proofs appear to proceed by case analysis on possible groups G containing such subgroups, which is the natural way to handle the cyclic situation. The stress-test concern about uniformity across non-conjugate H is addressed in the setup, since the claim is stated for arbitrary parallel extensions; if the case analysis really enumerates all possibilities without extra assumptions on the action of H, the result holds. A minor soft spot is that the paper is scoped tightly to cyclic N and p-powers, so it does not speak to other types or degrees where the same question remains open. This is for people already working on Hopf-Galois classifications or Galois module theory for p-power extensions. A reader who needs the cyclic case settled will get immediate use from the explicit group-theoretic conditions. It deserves a serious referee because the reduction is standard and the claim is precise enough that the details can be verified without broad new theory. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. The paper claims to completely solve the question of whether the existence of a Hopf-Galois structure of type N on a finite separable extension L/K of prime power degree implies the same for every parallel extension L'/K of equal degree. It reduces the problem via the Greither-Pareigis and Byott correspondences to a purely group-theoretic question: for G = Gal(tilde L / K) and H ≤ G with [G:H] = p^k corresponding to L, does there exist T ≤ G with T ≅ N (cyclic of order p^k), T ∩ H = {1}, and TH = G? The manuscript asserts a positive answer that holds uniformly for all such parallel extensions.

Significance. If correct, the result gives a definitive group-theoretic resolution for the cyclic case of prime-power degree, clarifying the behavior of Hopf-Galois structures under the parallel-extension relation. The approach correctly invokes the established Greither-Pareigis/Byott dictionary and focuses on the non-trivial uniformity requirement across all core-free subgroups of index p^k rather than a single extension. This supplies a clean base case that future work on non-cyclic types or composite degrees can build upon.

major comments (1)

The central claim of a 'complete solution' is load-bearing on uniformity: existence of T for one core-free H of index p^k must imply existence of (possibly different) T' for every other core-free H' of the same index, since parallel extensions correspond to arbitrary such subgroups, not merely conjugates. The manuscript should state explicitly (in the proof of the main theorem) whether the case analysis for cyclic N uses only properties that are invariant under choice of H or whether additional verification is supplied for arbitrary core-free subgroups of index p^k.

minor comments (1)

The introduction would benefit from a brief concrete example (e.g., a degree-p^2 extension with two distinct parallel subfields) to illustrate the distinction between conjugate and non-conjugate core-free subgroups of equal index.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading of the manuscript and for highlighting the importance of uniformity in the main result. We address the single major comment below and agree to strengthen the exposition as suggested.

read point-by-point responses

Referee: The central claim of a 'complete solution' is load-bearing on uniformity: existence of T for one core-free H of index p^k must imply existence of (possibly different) T' for every other core-free H' of the same index, since parallel extensions correspond to arbitrary such subgroups, not merely conjugates. The manuscript should state explicitly (in the proof of the main theorem) whether the case analysis for cyclic N uses only properties that are invariant under choice of H or whether additional verification is supplied for arbitrary core-free subgroups of index p^k.

Authors: We agree that explicit clarification on this point will improve the readability of the argument. The proof of the main theorem (Theorem 4.1) proceeds via a case analysis on the possible isomorphism types of the Galois group G of prime-power order. For each case, the existence of a complement T ≅ N (cyclic of order p^k) to a core-free H is established using only the order of G, the fact that N is cyclic, and the general fact that any two core-free subgroups of index p^k in such a G are related by the same numerical invariants (e.g., the structure of the Frattini quotient and the possible actions). These properties are manifestly independent of the particular choice of H. Consequently the argument already applies uniformly to every core-free subgroup of index p^k. We will revise the manuscript by inserting, immediately before the case analysis in the proof of Theorem 4.1, a short paragraph stating that the subsequent reasoning relies exclusively on H-invariant data and therefore holds for an arbitrary core-free subgroup of the given index. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses external Greither-Pareigis/Byott correspondences with independent content

full rationale

The paper translates the Hopf-Galois question for parallel extensions to a group-theoretic existence problem for subgroups T ≅ N (cyclic of order p^k) with T ∩ H = {1} and TH = G, where H ≤ G = Gal(tilde L/K) has index p^k. This reduction is justified by citing the Greither-Pareigis and Byott correspondences, whose authors are distinct from the present ones. The complete solution for prime-power degree and cyclic N proceeds via case analysis on the group G without any self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or claims in the provided abstract or description reduce the central result to its inputs by construction. The derivation remains self-contained against the external benchmarks cited.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The solution rests on the Greither-Pareigis correspondence between Hopf-Galois structures and certain subgroups of the Galois group, plus Byott's reformulation; these are treated as standard background rather than new axioms.

axioms (1)

domain assumption The Greither-Pareigis and Byott correspondences between Hopf-Galois structures of type N and certain subgroups or homomorphisms in the Galois group of the normal closure hold for separable extensions of prime-power degree.
Invoked to reduce the field-theoretic question to group theory; cited in the abstract as the basis of the approach.

pith-pipeline@v0.9.0 · 5667 in / 1289 out tokens · 25568 ms · 2026-05-18T06:36:28.831934+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We completely solve this problem when the degree [L:K] is a prime power and the type N is cyclic. Our approach is group-theoretic and uses the work of Greither–Pareigis and Byott.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.4 ... odd prime power ... Theorem 1.5 ... even prime power ... Proposition 5.3, 5.4, 5.7, 5.10

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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W. Bosma, J. Cannon, and C. Playoust,The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235–265

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Darlington,Hopf–Galois structures on parallel extensions, J

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C. Greither, B. Pareigis,Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), no. 1, 239–258

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C. Tsang,The multiple holomorph of a semidirect product of groups having coprime exponents, Arch. Math. (Basel) 115 (2020), no. 1, 13–21. 31 A. Darlington,Department of Mathematics and Data Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium Emailandrew.darlington@vub.be Homepagehttps://sites.google.com/view/andrewdarlington/ C. Tsan...

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[1] [1]

Bosma, J

W. Bosma, J. Cannon, and C. Playoust,The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235–265

work page 1997

[2] [2]

N. P. Byott,Uniqueness of Hopf Galois structure for separable field exten- sions, Comm. Algebra 24 (1996), no. 10, 3217–3228. Corrigendum,ibid.no. 11, 3705

work page 1996

[3] [3]

Campedel, A

E. Campedel, A. Caranti, I. Del Corso,Hopf-Galois structures on exten- sions of degreep 2qand skew braces of orderp 2q: the cyclic Sylowp-subgroup case, J. Algebra 556 (2020), 1165–1210

work page 2020

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S. U. Chase, M. E. Sweedler,Hopf algebras and Galois theory, Lecture Notes in Mathematics, Vol. 97. Springer-Verlag, Berlin-New York, 1969

work page 1969

[5] [5]

L. N. Childs,Taming wild extensions: Hopf algebras and local Galois mod- ule theory, Mathematical Surveys and Monographs, 80. American Mathe- matical Society, Providence, RI, 2000

work page 2000

[6] [6]

Crespo,Automatic realization of Hopf Galois structures, J

T. Crespo,Automatic realization of Hopf Galois structures, J. Algebra Appl. 21 (2022), no. 2, Paper No. 2250030, 9 pp

work page 2022

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Darlington,Hopf–Galois structures on parallel extensions, J

A. Darlington,Hopf–Galois structures on parallel extensions, J. Algebra 679 (2025), 1–27

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Greither, B

C. Greither, B. Pareigis,Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), no. 1, 239–258

work page 1987

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Guarnieri, L

L. Guarnieri, L. Vendramin,Skew braces and the Yang-Baxter equation, Math. Comp. 86 (2017), no. 307, 2519–2534

work page 2017

[10] [10]

M. R. Murty, V. K. Murty,On groups of squarefree order, Math. Ann. 267 (1984), no. 3, 299–309

work page 1984

[11] [11]

Rump,Classification of cyclic braces, J

W. Rump,Classification of cyclic braces, J. Pure Appl. Algebra 209 (2007), no. 3, 671–685

work page 2007

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Suzuki,Group theory

M. Suzuki,Group theory. II, Translated from the Japanese. Grundlehren der mathematischen Wissenschaften, 248. Springer-Verlag, New York, 1986

work page 1986

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Tsang,The multiple holomorph of a semidirect product of groups having coprime exponents, Arch

C. Tsang,The multiple holomorph of a semidirect product of groups having coprime exponents, Arch. Math. (Basel) 115 (2020), no. 1, 13–21. 31 A. Darlington,Department of Mathematics and Data Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium Emailandrew.darlington@vub.be Homepagehttps://sites.google.com/view/andrewdarlington/ C. Tsan...

work page 2020