All totally real abelian number fields of class number one are Euclidean

Harald Grobner

REVIEW

All totally real abelian number fields of class number one are Euclidean

Harald Grobner

Number Theory · Submitted 17 Oct 2026

Pith's verdict: Cleverly structured but rests entirely on two unverified black-box assumptions from an AI repository—the paper is a conditional theorem, not a proof.

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arXiv ID2610.24900 (version 2)

Pith IDTQKHXFNB, Pith's permanent record of this paper and its review

CitationHarald Grobner. "All totally real abelian number fields of class number one are Euclidean." arXiv:2610.24900 (2026). https://pith.science/paper/TQKHXFNB

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Editor's note

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Harald has written a clean, well-organized note that reduces the remaining cases of Narkiewicz's classification to two analytic assumptions from an OpenAI preprint. The internal logic is sound: the descent argument (Lemma 1.4), the refined zero-counting (Proposition 1.6), and the handling of the exceptional divisors (Sections 2.2–2.3) are all executed with care. The paper does what it says on the tin—if [OAI26] holds, the Euclidean ⇔ h_K = 1 statement follows for all totally real abelian fields.

The novelty is real: the lower bound for Π_K(x) in degrees 2 and 3 was previously open, and Harper–Murty plus Murty–Petersen left those cases conditional on GRH or additional Galois conditions. Grobner's framework is a serious attempt to close them unconditionally.

The problem, as the reader and the stress test both correctly flag, is that the two Scholia from [OAI26] are load-bearing and unverified. Scholium (II) in particular is a bespoke prime-counting estimate that is far from standard; the paper offers no proof, only a citation to an unreviewed repository. The entire chain of inequalities in §3 depends on a positive proportion of primes satisfying that very specific factorization condition. If the estimate fails—even in the constant a_k or the threshold x_k—the main result collapses. The paper acknowledges the reliance transparently, but that does not reduce the risk.

My take: this is a plausible conditional theorem. It deserves a serious referee, because the internal mathematics is careful and the conclusion is important. But the referee would need to evaluate [OAI26] as well, and that preprint's provenance (OpenAI mathematics repository, no conventional peer review) is a serious obstacle. As of now, I would not cite this result in my own work, and I wouldn't bring it to a reading group until the assumptions are independently verified. The paper itself is not at fault, but the epistemic status is provisional.

Referee report

Major objections (2)

[Introduction; Scholia (I) and (II)] The paper's central claim (Main Theorem) depends entirely on Scholia (I) and (II), which are cited from the preprint [OAI26] without proof. Since [OAI26] is an unpublished preprint and its results have not been verified by the community, the paper does not provide a valid unconditional proof. This is a load-bearing deficiency that cannot be fixed within the current manuscript; the paper would need either to include full proofs of these Scholia or to cite a peer-reviewed source.
[Scholium (II); §2.1] Scholium (II) asserts that for every integer k ≥ 1 there exist constants a_k, x_k > 0 such that for all x ≥ x_k, at least a_k x / (log x)^2 primes p in (x,2x) satisfy p ≡ −1 (mod 8k) and p−1 = 2^r q with q > x^{9/10} prime and every prime divisor of r in [exp(log^{1/10} x), exp(log^{3/10} x)]. This is an extremely strong and unnatural prime-counting estimate. The paper provides no heuristic, numerical evidence, or proof. If this estimate is false or requires astronomically large x_k, the lower bound for Π_K(x) and consequently the entire Euclidean ⇔ h_K=1 implication collapses. The paper's own estimates for A(x) and B(x) in §2.2 and §2.3 are premised on the existence of a positive proportion of such primes.

Minor objections (4)

[References; Introduction] The reference [OAI26] is attributed to "OpenAI, Primitive roots for every admissible integer base, preprint, Oct 4, (2026) OpenAI mathematics repository." This is an unconventional source for a mathematical proof. The authors should clarify the status of this preprint and, ideally, cite a refereed publication or provide the results as an appendix.
[§1.3, proof of Proposition 1.6] The argument constructing T_j and bounding the horizontal integrals (paragraphs following (1.16)) depends on a field-dependent constant C_F. For the application in Corollary 1.20 to H_ℓ (which varies with ℓ), the implied constant in (1.7) must be uniform in ℓ. The proof as written establishes (1.7) with an implied constant that depends on F; the deduction that the same constant works for all H_ℓ with ℓ ≤ exp(log x^{3/10}) requires additional justification or an explicit discussion of uniformity.
[§1.2, Lemma 1.2] The bound for log D_{H_ℓ} / [H_ℓ:Q] is given as ≤ (1/n) log D + 2 log ℓ. The derivation uses (2ℓ−3)/(ℓ−1) < 2, which is valid for ℓ ≥ 2. Since ℓ is an odd prime, ℓ ≥ 3 holds, so the inequality is correct, but the statement could be made clearer by noting this range.
[§2.2, equation (2.3)] The sum over primes ℓ is approximated by the sum over integers n ≥ ⌈exp(log^{1/10} x)⌉ of 1/(n(n−1)). The error introduced by replacing primes by integers is not discussed. While this does not affect the o(·) conclusion, a brief justification would improve rigor.

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The paper

The paper claims

All class-number-one totally real fields shown Euclidean, conditional on two conjectures

The reading

This paper tries to prove that for every totally real abelian number field $K/\mathbb{Q}$ with $[K:\mathbb{Q}]\ge 2$, the ring of integers $\mathcal{O}_K$ is a Euclidean domain if and only if the class number $h_K = 1$. This includes the case of real quadratic and totally real cubic fields, where Narkiewicz had left open the possibility of up to two quadratic and one cubic exceptional fields that could be UFDs without being Euclidean. The proof builds on two powerful but currently unverified assertions from the preprint [OAI26]: a uniform zero-free region of width $10^{-6}$ for Hecke $L$-functions of cyclotomic fields containing $\mu_{12}$, and a lower bound on the count of primes with a special factorization. If these are true, the paper closes the gap completely and verifies a prediction of Weinberger.

What carries the argument

The argument rests on two unproven pillars from [OAI26]: (I) a uniform zero-free region $\operatorname{Re}(s) > 1 - \delta_0$ with $\delta_0 = 10^{-6}$ for all Hecke $L$-functions $L(s,\chi)$ of finite order for every cyclotomic field that contains the $12$th roots of unity; (II) a lower bound $\gg x/(\log x)^2$ on the number of rational primes $p$ in $(x,2x)$ satisfying $p\equiv -1\pmod{8k}$ and $p-1 = 2rq$ with $q > x^{9/10}$ prime and all prime divisors of $r$ between $\exp(\log(x)^{1/10})$ and $\exp(\log(x)^{3/10})$. Additionally, the author proves a general estimate for $\Theta_{F,\varphi}(x)$ (Proposition 1.6), which bounds the weighted sum over prime ideal powers in terms of $x\hat{\varphi}(1) + x^{\Theta_F}(\log D_F + [F:\mathbb{Q}])$, and uses it together with the two Scholia to derive the required bound on $\Pi_K(x)$.

Extended reading notes

Core claim

Assuming the two black-box statements from [OAI26] (Scholia (I) and (II)), the paper establishes a lower bound $\Pi_K(x) \ge c_K \, x / (\log x)^2$ for the number of prime ideals of $K$ whose norm is prime and lie in $(x,2x)$ and for which the natural map $\mathcal{O}_K^* \to (\mathcal{O}_K/\mathfrak{p})^*$ is surjective. By Harper–Murty's criterion, this lower bound implies that $\mathcal{O}_K$ is Euclidean whenever $h_K = 1$. The technical heart is a new estimate for the weighted sum $\Theta_{F,\varphi}(x)$ (Proposition 1.6), which the author proves unconditionally for any number field $F$, and which, combined with the zero-free region, yields control over the prime ideals that split in certain ray class fields. The paper's novelty concentrates on the previously open degrees $[K:\mathbb{Q}]=2,3$.

Load-bearing premise

The proof assumes that two unproven number-theoretic statements are true: (1) There exists a uniform zero-free region of width $10^{-6}$ for all Hecke $L$-functions of cyclotomic fields containing the $12$th roots of unity; (2) For every integer $k$, there are at least $a_k x/(\log x)^2$ primes $p$ in $(x,2x)$ with $p \equiv -1 \pmod{8k}$ and $p-1 = 2rq$ where $q > x^{9/10}$ is prime and all prime divisors of $r$ lie in a prescribed interval. If either assumption fails, the main theorem is unsupported.

Editorial extensions

If this is right

If the two conjectures from [OAI26] are eventually verified, then for all totally real abelian number fields $K/\mathbb{Q}$ with $[K:\mathbb{Q}]\ge 2$, the condition $h_K = 1$ is equivalent to $\mathcal{O}_K$ being Euclidean — closing Narkiewicz's 2007 list of possible non-Euclidean UFDs.
The result would confirm Weinberger's prediction that the exceptional real quadratic and cubic fields should not exist without assuming the full generalized Riemann hypothesis; only the weaker Quasi-Riemann hypothesis (uniform zero-free region) is needed.
The unconditional Proposition 1.6 of the paper provides a new tool for bounding prime ideal sums in arbitrary number fields, which may be useful in other analytic number theory contexts beyond Euclideanity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's entire edifice is conditional on two statements from an unreviewed source. Until the Scholia are independently proved in a refereed publication, the main theorem remains a conditional result, not an unconditional theorem.
The author's Proposition 1.6 improves on [OAI26, Lem. 9.2] by making the implied constant independent of the field and gives a complete proof. This technical contribution may survive even if the Scholia are eventually found to be false, as it requires only general assumptions about the Riemann constant $\Theta_F$.
If the zero-free region (Scholium I) could be replaced by a weaker statement applicable to the specific fields $H_\ell$, or if the prime-counting lemma (Scholium II) could be established by classical sieving methods, the result might become unconditional without the full strength of the Quasi-Riemann hypothesis.

Original abstract

It is a famous result of Narkiewicz that there are at most two real quadratic and at most one totally real cubic Galois extensions $K/\mathbb Q$, whose ring of integers $\mathcal O_K$ is a unique factorization domain (UFD), but not Euclidean. As predicted by another famous result due to Weinberger, these exceptions should in fact not exist, as their existence would contradict a generalized version of the Riemann hypothesis. Building on recent spectacular progress on the Quasi-Riemann hypothesis for Hecke characters, in this short note we close the above final gap provided by Narkiewicz's result, and show, more generally, that the ring of integers $\mathcal O_K$ of any totally real abelian number field $K$ is a UFD, if and only if it is Euclidean. By referring to an additional announced uniform zero-free half-plane for general Dedekind zeta functions, we also establish the corresponding result for certain explicit new families of non-Galois cubic, quartic and quintic fields of unit rank one or two.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 3,176 input tokens.

What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

It is a famous result of Narkiewicz that there are at most two real quadratic and at most one totally real cubic Galois extensions $K/\mathbb Q$, whose ring of integers $\mathcal O_K$ is a unique factorization domain (UFD), but not Euclidean. As predicted by another famous result due to Weinberger, these exceptions should in fact not exist, as their existence would contradict a generalized version of the Riemann hypothesis. Building on recent spectacular progress on the Quasi-Riemann hypothesis for Hecke characters, in this short note we close the above final gap provided by Narkiewicz's result, and show, more generally, that the ring of integers $\mathcal O_K$ of any totally real abelian number field $K$ is a UFD, if and only if it is Euclidean.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	It is a famous result of Narkiewicz that there are at most two real quadratic and at most one totally real cubic Galois extensions $K/\mathbb Q$, whose ring of integers $\mathcal O_K$ is a unique factorization domain (UFD), but not Euclidean.	background 0.93	0.02	0.03	asserted 1.00
s1	As predicted by another famous result due to Weinberger, these exceptions should in fact not exist, as their existence would contradict a generalized version of the Riemann hypothesis.	background 0.96	0.03	0.03	hedged possibility 0.69
s2	Building on recent spectacular progress on the Quasi-Riemann hypothesis for Hecke characters, in this short note we close the above final gap provided by Narkiewicz's result, and show, more generally, that the ring of integers $\mathcal O_K$ of any totally real abelian number field $K$ is a UFD, if and only if it is Euclidean.	contribution 0.56	0.95	0.97	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Discussion

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