A Dedekind's Criterion over Valued Fields

Abdulaziz Deajim; Lhoussain El fadil; Mhammed Boulagouaz

REVIEW 2 major objections 5 minor 8 references

A Dedekind's Criterion over Valued Fields

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Over arbitrary-rank valued fields, integral closedness of a simple extension ring reduces to a finite remainder check.

desk verdict A genuinely useful generalization of Dedekind's criterion to arbitrary-rank valued fields, but the proof of the main sufficiency direction has a fixable gap in the lift-adjustment step. read the letter →

arxiv 1908.06365 v1 pith:I2WMX5XK submitted 2019-08-18 math.NT

Lhoussain El Fadil , Mhammed Boulagouaz , Abdulaziz Deajim This is my paper · ORCID

classification math.NT MSC 12J1013A1813B22

keywords Dedekind'scriterionvaluedfieldextensionsofavaluationintegralclosureGaussianHenselizationramificationindexEisensteinpolynomial

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper gives necessary and sufficient conditions, for an arbitrary-rank valued field $(K,\nu)$, under which the subring $R_\nu[\alpha]$ of a finite separable extension $K(\alpha)$ equals the full integral closure of the valuation ring. The answer is read from the reduction of $f$ modulo the maximal ideal: if $\bar f = \prod \bar\varphi_i^{\ell_i}$, then $R_\nu[\alpha]$ is integrally closed exactly when, for every repeated factor, the remainder of $f$ upon Euclidean division by a lift $\varphi_i$ has Gaussian valuation equal to the smallest positive element of the value group. If no smallest positive value exists, integral closedness holds precisely in the square-free case. The same condition is rephrased in terms of the valuations on $K(\alpha)$ extending $\nu$, and it yields the ramification indices and residue degrees of all those extensions.

What carries the argument

The load-bearing object is the pair consisting of the Gaussian valuation $\nu_G$ and the remainder $r_i$: $\nu_G$ takes the minimum of the $\nu$-values of the coefficients of a polynomial, and $r_i$ is the remainder of $f$ modulo the lifted irreducible factor $\varphi_i$. The criterion's condition $\nu_G(r_i)=\min(\Gamma_\nu^+)$ is a finite, checkable measurement of how far $f$ is from being a product of powers of the $\varphi_i$ modulo the maximal ideal. The other central mechanism is the Henselization correspondence of Lemma 2.1: valuations on $K(\alpha)$ extending $\nu$ biject with irreducible factors of $f$ over the Henselization $K^h$, allowing each extension to be identified by the unique residue factor $\varphi_i$ that becomes positive at $\varphi_i(\alpha)$.

What would settle it

For $K=F(X,Y)$ with the lexicographic valuation on $\mathbb{Z}^2$ (where $\min(\Gamma_\nu^+)=(0,1)$) and $f(Z)=Z^3+YZ+X$, the theorem predicts $R_\nu[\alpha]$ is not integrally closed because the remainder of $f$ modulo $Z$ is $r=X$ and $\nu_G(r)=(1,0)\neq(0,1)$. Directly computing the integral closure of $R_\nu[Z]/(f)$ in this case, and either exhibiting an element of $S\setminus R_\nu[\alpha]$ or finding $S=R_\nu[\alpha]$, would settle the criterion on a concrete instance.

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Extended reading notes

Core claim

Under the paper's standing assumptions, Theorem 2.5 states: if $I = \{\,i \mid \ell_i \ge 2\,\}$ is nonempty, then $R_\nu[\alpha]$ is integrally closed if and only if $\nu_G(r_i) = \min(\Gamma_\nu^+)$ for every $i \in I$, where $r_i$ is the remainder on Euclidean division of $f$ by the monic lift $\varphi_i$ of the $i$-th irreducible residue factor, and $\nu_G$ is the Gaussian extension of $\nu$ to $K[X]$. Theorem 2.9 gives an equivalent valuation-extension form: integral closedness holds exactly when $\nu$ has $s$ distinct extensions to $K(\alpha)$, one for each residue factor, and for each repeated factor the value $\ell_i\,\omega_i(\varphi_i(\alpha))$ is the minimum positive element of $\Gamma_\nu$. When this happens, Corollary 2.10 computes $e(\omega_i/\nu)=\ell_i$ and $f(\omega_i/\nu)=\deg(\varphi_i)$, and the fundamental inequality becomes an equality.

Load-bearing premise

The argument rests on a bijection, supplied by Lemma 2.1, between valuations on $K(\alpha)$ extending $\nu$ and irreducible factors of $f$ over the Henselization $K^h$; this bijection is only guaranteed for separable $f$, so if $f$ fails to be separable the whole characterization loses its footing.

Editorial extensions

If this is right

If $\Gamma_\nu^+$ has a least element $\sigma$, the criterion reduces to checking finitely many remainders: $R_\nu[\alpha]$ is integrally closed if and only if every repeated-factor remainder has Gaussian value exactly $\sigma$.
If $\Gamma_\nu^+$ has no least element, then integral closedness can occur only in the square-free case $\ell_i=1$ for all $i$; any repeated residue factor forces $R_\nu[\alpha]$ to be non-closed.
When the criterion holds, $\nu$ splits into exactly $s$ extensions, with ramification index $\ell_i$ and residue degree $\deg(\varphi_i)$ attached to the $i$-th residue factor, so the fundamental inequality becomes an equality.
For a pure extension $X^n-a$ with $a \in M_\nu$ and $\min(\Gamma_\nu^+)=\sigma$, integral closedness of $R_\nu[\alpha]$ is equivalent to $\nu(a)=\sigma$; in the coprime case the integral closure is exhibited explicitly as $R_\nu[\theta^v/\pi^u]$.

Reading between the lines

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

The criterion is computational in practice: for an explicit valued field with decidable value-group order, one can test integral closedness by Euclidean division and Gaussian valuation alone, without constructing the integral closure.
The paper's proof depends on separability through the Henselization correspondence, so a natural stress test is to search for a non-separable $f$ where the remainder condition holds but integral closedness fails; such an example would mark the theorem's boundary.
The equality $e(\omega_i/\nu)=\ell_i$, $f(\omega_i/\nu)=\deg(\varphi_i)$ suggests that integrally closed orders are exactly those that realize the full splitting of $\nu$ predicted by the residue factorization, which could guide constructions of extensions with prescribed ramification data.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read this paper closely, and I think you should know two things up front. First, it does contain new and citable results: Theorem 2.9, characterizing integral closedness of Rν[α] in terms of s distinct extensions and the equality liω_i(φ_i(α)) = min(Γ+ν), and Corollary 2.10, computing ramification indices and residue degrees as li and deg(φ_i) when the order is integrally closed. These are genuine advances over the Khanduja–Kumar framework, and the paper correctly notes that the Henselian hypothesis is not needed. Second, the proof of Theorem 2.5(ii) has a real gap, though I think it is repairable.

The main problem is in the paragraph where the authors claim that for i not in I with νG(ri) > σ, setting φ_i** = φ_i + π, q_i** = q_i − πq_i*, and r_i** = r_i − πr_i* + π^2q_i* gives the Euclidean quotient and remainder of f by φ_i**. That claim is false in general. The stress-test example is correct: for K = Q with the 5-adic valuation and f = X^3 + X^2 + 25, reducing mod 5 gives X^2(X+1), so for the simple factor φ_i = X+1 we get q_i* = X−1, r_i* = 1, and then r_i** = 25X−5 has degree 1, equal to deg(φ_i**), so it cannot be the remainder. The actual remainder of f by X+6 is −155.

There is also a separate issue in the final contradiction argument. The authors write that ω(g(α)) = miσ/li < σ, but if the chosen i has mi = 0 (which can happen for a simple factor, or even for a repeated factor when φ_i does not divide g), then miσ/li = 0, which is not less than σ. The conclusion still follows if one splits off the mi = 0 case: then φ_i does not divide g, so by Lemma 2.2(iv) ω(g(α)) = 0, and since ν(b) ≥ σ, ω(θ) ≤ −σ < 0. So the theorem itself appears true, but the written proof is not complete.

Other soft spots are minor: Lemma 2.3's phrase "r_i is zero" should read "r_i maps to zero in kν[X]", and the introduction has garbled sentences. The citation pattern is fine; the self-citation [1] is not problematic because Lemma 2.2 is proved in full. Separability is explicitly assumed, which is appropriate.

My recommendation: this paper deserves serious peer review, but the referee should ask for a corrected proof of Theorem 2.5(ii) that either fixes the lift-adjustment step or removes it by handling mi = 0 separately. The new results are substantial enough that a revision would be worth engaging with.

Referee Report

2 major / 5 minor

Summary. The paper proves a generalized Dedekind criterion for integral closedness of Rν[α] in a separable finite simple extension L=K(α) of an arbitrary-rank valued field (K,ν). Under Assump's (f monic irreducible separable in Rν[X], with reduction factorized as ∏ φ_i^{l_i}), Theorem 2.5 characterizes integral closedness by the condition νG(r_i)=min(Γ_ν^+) for every i with l_i≥2, where r_i is the remainder of f modulo φ_i. Theorem 2.9 gives an equivalent characterization in terms of the number of extensions of ν to L and the values l_iω_i(φ_i(α)). The paper also derives ramification and residue-degree data (Corollary 2.10), an Eisenstein-style irreducibility lemma, and applications (Corollaries 3.1–3.3, Examples 1–2).

Significance. If the results are correct, the paper gives a clean and useful generalization of Ershov's and Khanduja–Kumar's Dedekind criteria to arbitrary-rank valued fields, removing the Henselian assumption, and it computes the ramification and residue-degree decomposition in the integrally closed case. The manuscript is largely self-contained: Lemma 2.2, generalizing a previous result of the authors, is proved in full, and Lemma 2.4 contains a sharp and correct identity l_iω(φ_i(α))=σ. The main theorems are falsifiable and illustrated with concrete examples. However, the proof of the sufficiency direction of Theorem 2.5 contains two gaps that need repair before the result can be accepted.

major comments (2)

[Section 2, proof of Theorem 2.5(ii), paragraph beginning 'By making an appropriate choice of a lifting...'] The claim that the polynomials q_i** and r_i** defined there are the Euclidean quotient and remainder of f by φ_i** is not valid in general. For example, take K=Q with the 5-adic valuation (π=5, σ=1), f(X)=X^3+X^2+25, φ_i=X+1. Then q_i=X^2, q_i*=X−1, r_i*=1, φ_i**=X+6, q_i**=X^2−5X+5, and r_i**=25X−5. Here deg(r_i**)=deg(φ_i**)=1, so r_i** is not the remainder; the true remainder is −155. Since this step is used to justify reducing to the case νG(r_i)=σ for all i, and hence to apply Lemma 2.4 to every i in the final argument, the proof as written is incomplete. The gap is repairable: for i∉I one has l_i=1, and in the final contradiction one may take m_i=0, so the precise value of ω(φ_i(α)) is not needed; alternatively one must prove that the true remainder after further Euclidean division still has Gaussian value σ.
[Section 2, proof of Theorem 2.5(ii), final paragraph] The assertion that after writing g = S_i φ_i^{m_i} + T_i with φ_i ∤ S_i one has νG(T_i) ≥ σ is not a consequence of the definition of m_i. If m_i is the highest power of φ_i dividing g in kν[X], the remainder T_i may have a coefficient of value 0, so in general only νG(T_i) ≥ 0 holds. The subsequent equality ω(g(α)) = m_iσ/l_i relies on the stronger bound and is therefore not established. The desired contradiction can still be recovered: since ω(T_i(α)) ≥ νG(T_i) ≥ 0 and m_iσ/l_i ≥ 0, the minimum is 0 < σ, giving ω(g(α)) = 0 < σ and hence ω(θ) < 0. The proof should be corrected to use this weaker but sufficient bound.

minor comments (5)

[Lemma 2.3 and Corollary 2.7] The phrase 'r_i is zero' is used where the argument only shows that the reduction of r_i modulo Mν is zero (equivalently νG(r_i)>0). This should be reworded to avoid a literal false statement.
[Page 1, after the abstract] The sentence 'on does not explicitly state the assumption that f is separable, both references crucially use, which in turn assumes the separability of f' is garbled and needs to be rewritten.
[Theorem 2.5(i), proof] In the line 'deg(h) ≤ deg(h)', the notation should clarify that h denotes the reduction of h modulo Mν.
[Corollary 3.3] The notation 'R[θv/π u]' should be typeset as Rν[θ^v/π^u], and 'A = av/πnu' as A = a^v/π^{nu}.
[References] Reference [1] is listed as 'to appear'; if the paper is still unpublished, please update the status or provide a preprint reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation is self-contained, with the sole self-citation not load-bearing.

full rationale

The paper's derivation chain is self-contained. Theorem 2.5 is proved from Lemmas 2.1-2.4: Lemma 2.1 is quoted from the external reference Endler, Lemma 2.2 is proved in full in the paper, and Lemmas 2.3-2.4 are new arguments built on those lemmas. The direct and converse directions do not assume the target conclusion. Theorem 2.9 is derived from Theorem 2.5, Lemma 2.1, and Lemma 2.8, none of which is equivalent to the conclusion. The only self-citation is the attribution 'The following result is a generalization of [1, Lemm 2.1]' before Lemma 2.2, but Lemma 2.2 is not imported from that citation: its proof is given in the paper and depends only on Lemma 2.1 and the stated assumptions. No uniqueness theorem is imported from the authors' own prior work, and no fitted parameter is renamed as a prediction. The definition of a nu-Eisenstein polynomial repackages the numerical condition appearing in Theorem 2.5, but it is not used as an input to the main criterion; it is applied after Theorem 2.5 is established. The alleged invalid Euclidean-division step in the proof of Theorem 2.5(ii) is a potential correctness gap, not a circularity: even if it is repairable, it does not show that the theorem is equivalent to its assumptions by construction. Accordingly, the paper exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters and no new entities. It rests on standard structural results of valuation theory: the correspondence between extensions and factors over the Henselization, the intersection description of the integral closure, Hensel's Lemma, the fundamental inequality, and Gauss's Lemma. The one domain assumption is separability of f, which is stated in the setup and is load-bearing for Lemma 2.1 and Lemma 2.2.

assumptions (6)

domain assumption Separability of f and of K(α)/K
Stated in Assump's premise; the proof of Lemma 2.2 and the use of Endler's [3, 17.17] in Lemma 2.1 require f separable. The authors explicitly note that [7] omits this hypothesis.
standard math Hensel's Lemma factorizes f over the Henselization K^h into f = ∏ f_i with f_i ≡ φ_i^{l_i} mod M_{ν^h}
Used in Lemma 2.2(i) and Theorem 2.9 to relate extensions to residue factors; a standard theorem for Henselian fields.
standard math Endler [3, 17.17] correspondence between extensions of ν to L and irreducible factors of f over K^h
Lemma 2.1, used throughout Theorems 2.5 and 2.9, gives the bijection ω_j ↔ f_j and the value formula.
standard math Integral closure S equals the intersection of all valuation rings R_ω for extensions ω of ν to L
Used in Lemma 2.2(ii) and Lemma 2.3 via [4, Corollary 3.1.4].
standard math Fundamental inequality ∑ e_i f_i ≤ [L:K] for extensions of valuations
Invoked in Corollary 2.10 to turn the computed lower bounds into equalities.
standard math Gauss's Lemma: irreducibility over the fraction field follows from irreducibility over the integrally closed valuation ring
Used in Lemma 2.8 to pass from irreducibility over Rν to irreducibility over K.

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Cite this review

Pith. "Pith review of A Dedekind's Criterion over Valued Fields." pith.science (2026). https://pith.science/paper/I2WMX5XK

@misc{pith2026190806365,
  author       = {Pith},
  title        = {Pith review of: A Dedekind's Criterion over Valued Fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I2WMX5XK}},
  note         = {Machine review of arXiv:1908.06365}
}

abstract

Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient conditions for $R_\nu[\alpha]$ to be integrally closed. We further characterize the integral closedness of $R_\nu[\alpha]$ based on information about the valuations on $K(\alpha)$ extending $\nu$. Our results enhance and generalize some existing results in the relevant literature. Some applications and examples are also given.

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Deajim and L

A. Deajim and L. El Fadil, On the integral closedness of R[α ], to appear in Math. Reports

work page
[2]

Dedekind, Uber den zussamenhang zwischen der theorie der ideals und de r theorie der hoheren cyclotimy index , Abh

R. Dedekind, Uber den zussamenhang zwischen der theorie der ideals und de r theorie der hoheren cyclotimy index , Abh. Akad. Wiss. Gottingen, Math.-Phys. KL 23 (1878) 1–23

work page
[3]

Endler, Valuation Theory, Springer-Verlag, Berlin, 1972

O. Endler, Valuation Theory, Springer-Verlag, Berlin, 1972

work page 1972
[4]

Engler and A

A. Engler and A. Prestel, Valued Fields, Springer-Verlag, Berlin, 2005

work page 2005
[5]

Ershov, A Dedeking criterion for arbitrary valuation rings , Doklady Math 74 (2006), 650–652

Y. Ershov, A Dedeking criterion for arbitrary valuation rings , Doklady Math 74 (2006), 650–652

work page 2006
[6]

Khanduja and M

S. Khanduja and M. Kumar, A generalization of Dedekind criterion , Commun. in Algebra 35 (2007), 1479–1486

work page 2007
[7]

Khanduja and M

S. Khanduja and M. Kumar, On Dedekind criterion and simple extensions of valuation ri ngs, Com- mun. in Algebra 38 (2010), 684–696

work page 2010
[8]

Neukirch, Algebraic Number Theory , Springer-Verlag, 1999

J. Neukirch, Algebraic Number Theory , Springer-Verlag, 1999. 10 LHOUSSAIN EL F ADIL, MHAMMED BOULAGOUAZ, AND ABDULAZIZ DE AJIM (L. El Fadil) Department of Mathematics, F aculty of Sciences Dhar-Mahra z, University of Sidi Mohamed Ben Abdellah, B.P. 1796, Fes, Morocco E-mail address : lhouelfadil2@gmail.com (M. Boulagouaz) Department of Mathematics, F acu...

work page 1999

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Deajim and L

A. Deajim and L. El Fadil, On the integral closedness of R[α ], to appear in Math. Reports

work page

[2] [2]

Dedekind, Uber den zussamenhang zwischen der theorie der ideals und de r theorie der hoheren cyclotimy index , Abh

R. Dedekind, Uber den zussamenhang zwischen der theorie der ideals und de r theorie der hoheren cyclotimy index , Abh. Akad. Wiss. Gottingen, Math.-Phys. KL 23 (1878) 1–23

work page

[3] [3]

Endler, Valuation Theory, Springer-Verlag, Berlin, 1972

O. Endler, Valuation Theory, Springer-Verlag, Berlin, 1972

work page 1972

[4] [4]

Engler and A

A. Engler and A. Prestel, Valued Fields, Springer-Verlag, Berlin, 2005

work page 2005

[5] [5]

Ershov, A Dedeking criterion for arbitrary valuation rings , Doklady Math 74 (2006), 650–652

Y. Ershov, A Dedeking criterion for arbitrary valuation rings , Doklady Math 74 (2006), 650–652

work page 2006

[6] [6]

Khanduja and M

S. Khanduja and M. Kumar, A generalization of Dedekind criterion , Commun. in Algebra 35 (2007), 1479–1486

work page 2007

[7] [7]

Khanduja and M

S. Khanduja and M. Kumar, On Dedekind criterion and simple extensions of valuation ri ngs, Com- mun. in Algebra 38 (2010), 684–696

work page 2010

[8] [8]

Neukirch, Algebraic Number Theory , Springer-Verlag, 1999

J. Neukirch, Algebraic Number Theory , Springer-Verlag, 1999. 10 LHOUSSAIN EL F ADIL, MHAMMED BOULAGOUAZ, AND ABDULAZIZ DE AJIM (L. El Fadil) Department of Mathematics, F aculty of Sciences Dhar-Mahra z, University of Sidi Mohamed Ben Abdellah, B.P. 1796, Fes, Morocco E-mail address : lhouelfadil2@gmail.com (M. Boulagouaz) Department of Mathematics, F acu...

work page 1999