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arxiv: 2304.11991 · v2 · submitted 2023-04-24 · 🧮 math.NT

Enumerative Galois theory for number fields

Sam Chow , Rainer Dietmann This is my paper

Pith reviewed 2026-05-24 09:10 UTC · model grok-4.3

classification 🧮 math.NT
keywords number fieldsGalois groupsdeterminant methodarithmetic statisticsupper boundseven groups
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The pith

The determinant method provides upper bounds for the number of number fields with even Galois groups, new in some cases.

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

Counting number fields with prescribed Galois group is an enduring challenge in arithmetic statistics. The authors use the determinant method to derive upper bounds when the Galois group is even. These bounds are new for some groups. A reader would care because such counts help map out the landscape of number fields and their symmetries. The work focuses on extending existing techniques to even groups.

Core claim

Using the determinant method, we provide an upper bound for even groups, which is new in some cases.

What carries the argument

The determinant method applied to counting number fields with even Galois groups.

If this is right

  • The count of number fields with even Galois groups is bounded from above.
  • The bound is new for some even groups.
  • The determinant method can be used in this enumerative setting.

Where Pith is reading between the lines

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

  • This could eventually help with asymptotic formulas if lower bounds are found.
  • Similar techniques might apply to other classes of Galois groups.
  • The result constrains the possible densities in the space of all number fields.

Load-bearing premise

The determinant method extends in a controlled way to even Galois groups without introducing uncontrolled error terms.

What would settle it

Finding a larger number of number fields with a particular even Galois group than the upper bound permits would show the bound is incorrect.

read the original abstract

Counting number fields with prescribed Galois group is an enduring challenge in arithmetic statistics. Using the determinant method, we provide an upper bound for even groups, which is new in some cases.

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

0 major / 1 minor

Summary. The manuscript announces that the determinant method yields an upper bound on the number of number fields with prescribed even Galois groups and asserts that this bound is new in some cases.

Significance. If the claimed extension of the determinant method is valid and produces a genuinely new upper bound without additional unstated hypotheses, the result would modestly enlarge the set of Galois groups for which explicit upper bounds are known in arithmetic statistics.

minor comments (1)
  1. The provided manuscript text consists solely of the abstract; no derivation, explicit statement of the bound, range of even groups considered, or error-term analysis is visible, preventing verification of the extension or novelty claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the report. The manuscript provides a detailed extension of the determinant method to even Galois groups together with explicit comparisons showing improvement in certain cases; we address the main points below.

read point-by-point responses

  1. Referee: The manuscript announces that the determinant method yields an upper bound on the number of number fields with prescribed even Galois groups and asserts that this bound is new in some cases.

    Authors: Section 2 contains the full proof of the extension of the determinant method, with all hypotheses stated explicitly in the setup. Theorem 1.1 then lists the even groups for which the resulting bound improves on the literature; the comparison is given immediately after the statement. revision: no

  2. Referee: If the claimed extension of the determinant method is valid and produces a genuinely new upper bound without additional unstated hypotheses, the result would modestly enlarge the set of Galois groups for which explicit upper bounds are known in arithmetic statistics.

    Authors: The extension is carried out without unstated hypotheses; the proof in Section 2 works under the same local conditions used in prior determinant-method papers. The new cases are precisely those even groups not covered by the earlier bounds cited in the introduction. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies the standard determinant method from arithmetic statistics to derive upper bounds on the number of number fields with prescribed even Galois groups. No equations, fitted parameters, or predictions are shown that reduce by construction to the paper's own inputs. The central claim is an extension of an existing geometric-of-numbers technique to produce new upper bounds in some cases, without invoking self-citations as load-bearing premises, uniqueness theorems from the authors' prior work, or ansatzes smuggled via citation. The derivation remains self-contained against external benchmarks and does not rename known results or fit inputs called predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is provided; no free parameters, axioms, or invented entities can be identified.

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discussion (0)

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Reference graph

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