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REVIEW 2 major objections 3 minor 44 references

A decisive Theorem (Un th\'eor\`eme d\'ecisif)

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Over a discrete field, every unramified finite-presentation algebra is étale and finite-dimensional.

desk verdict Clean constructive proof of a known theorem; two missing justifications that are easy to supply. read the letter →

arxiv 2506.07098 v1 pith:OBYBPCGA submitted 2025-06-08 math.AC

Henri Lombardi This is my paper
classification math.AC MSC 13C1013C1513C1114B2503F65
keywords nettealgebraunramifiedétaletraciallyKählerdifferentialsdiscretefieldfinitepresentationconstructive
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

Over a discrete field — one in which every element is effectively either zero or invertible — an algebra given by finitely many polynomial equations that has zero Kähler differentials (an unramified, or 'nette', algebra) is automatically finite-dimensional as a vector space and is étale in the strong sense that its discriminant is invertible. The paper proves this decisive theorem constructively, with no appeal to nonconstructive choice principles. The consequence is that for finitely presented algebras over discrete fields, the unramified condition is exactly the étale condition: both reduce to being a finite product of separable polynomial quotients. This matters because it turns a difficult-looking cohomological condition into an explicit, computable algebraic structure, and it completes the chain of definitions in constructive commutative algebra.

What carries the argument

The central object is the module of Kähler differentials $\Omega_{A/K}$, presented as the cokernel of the transposed Jacobian matrix of the defining polynomial system; netteness is exactly the vanishing $\Omega_{A/K}=0$. The argument is carried by two mechanisms: a localization argument in Theorem 4.5 that transplants the system into $L=K(Y_1,\dots,Y_r)$ over a transcendental extension, where an étale finite algebra must have invertible $f'(x)$ and hence cannot admit the nonzero $K$-derivation $\partial_1$ that partial differentiation provides; and a rank-reducing induction using idempotents, which relies on Lemma 4.1 to ensure the algebra is reduced and therefore decomposes as a product of smaller nette algebras. The machinery is constructive: all uses of linear algebra over the discrete field are explicit.

What would settle it

Find a discrete field $K$ and a finite-presentation $K$-algebra $A$ with $\Omega_{A/K}=0$ for which $A$ is not a finite-dimensional reduced étale algebra; for instance, a nonzero nilpotent element or a discriminant that is not invertible would disprove the theorem. A concrete classical test is the purely inseparable extension $K[X]/\langle X^p - t\rangle$ over $K=\mathbb{F}_p(t)$: it must fail netteness, and verifying $\Omega_{A/K}\neq 0$ here checks that the theorem excludes exactly such non-separable pieces.

Watch

Extended reading notes

Core claim

Theorem 4.4 is the decisive claim: over a discrete field $K$, every nette (unramified) $K$-algebra of finite presentation is strictly finite, meaning finite-dimensional as a $K$-vector space, and tracially étale, meaning $\operatorname{Disc}_{A/K}$ is invertible. Equivalently, a finite-presentation $K$-algebra over a discrete field is nette if and only if it is étale. The proof first shows that netteness forces the Noether dimension of the polynomial system to be $\le 0$ (Theorem 4.5), by localizing at a purely transcendental extension and deriving a contradiction from a nonzero partial derivative. It then uses the preliminary Lemma 4.1, asserting that a nette strictly finite algebra is reduced, to split the algebra into smaller pieces via idempotents and conclude by induction on the vector-space rank that it is a finite product of algebras $K[X]/\langle g_i\rangle$ with separable polynomials $g_i$.

Load-bearing premise

The load-bearing step is Lemma 4.1, stated in Section 4 without proof: a nette (unramified) strictly finite algebra over a discrete field has no nonzero nilpotent elements; the proof of Theorem 4.3, and therefore of the decisive Theorem 4.4, proceeds through this lemma rather than deriving it from the stronger Lemma 4.2 or from a cited source.

Editorial extensions

If this is right

  • If $A$ is a finite-presentation $K$-algebra over a discrete field with $\Omega_{A/K}=0$, then $A$ is isomorphic to a finite product $\prod_i K[X]/\langle g_i\rangle$ with each $g_i$ a unitary separable polynomial.
  • Consequently the general definition of an étale algebra (local trivialization of the Jacobian) coincides over a discrete field with the strong tracial definition (invertible discriminant), so the two notions are the same for finitely presented algebras.
  • Nette algebras over a discrete field have Noether dimension at most $0$: any polynomial system whose transposed Jacobian is surjective defines a zero-dimensional reduced object, with only finitely many geometric points.
  • Because the proof avoids nonconstructive principles, it yields an algorithm that, given a nette presentation, computes the separable polynomials and idempotents exhibiting the étale decomposition.

Reading between the lines

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

  • A natural next step, not pursued in the paper, is to read the induction in Theorem 4.3 as an algorithm that outputs a primitive element for any nette algebra over a discrete field, giving a constructive primitive-element theorem independent of field cardinality assumptions.
  • The footnote to Theorem 4.4 suggests the same 'nette implies étale' collapse should hold over reduced zero-dimensional rings via the local-global machinery; if true, the present field-level proof would become a springboard for a ring-level version with algorithmic content.
  • An immediate checkpoint for any reader is whether Lemma 4.2 from the companion preprint, which states that a free finite nette algebra over an arbitrary commutative ring is étale, implies Lemma 4.1 directly; the paper leaves the implication implicit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper (arXiv:2506.07098, in French) gives a constructive, elementary proof that every nette (unramified) algebra of finite presentation over a discrete field is strictly finite and etale in the strong tracial sense of Definition 1.1, i.e. its discriminant is invertible. Sections 1–3 collect standard material on etale algebras over discrete fields, Kähler differentials, and smooth/nette/etale finite-presentation algebras. Section 4 contains the main results: Theorem 4.3 characterizes strictly finite etale algebras over a discrete field; Lemma 4.1 states that a nette strictly finite algebra over a discrete field is reduced but is not proved; Theorem 4.4 asserts the decisive result, reducing it to Theorem 4.5, which claims that Ω_{A/K}=0 forces the Noether dimension of the system to be at most 0. The proof of Theorem 4.5 argues by contradiction with B=K[Y_1,...,Y_r] and C=S^{-1}A, and is meant to derive a contradiction from a nonzero derivation. The paper states that all proofs are constructive and implementable.

Significance. If completed, this note would provide a short, constructive proof of a known theorem: over a discrete field, finitely presented unramified algebras are finite-dimensional and tracially etale. The constructive decoding in Theorem 4.3 and the explicit use of Jacobian matrices are useful features, and the paper is honest about its relation to the authors' earlier book and to the companion preprint [1]. However, the main theorem is not new, and the two gaps identified below affect the proof of the central claim, so the significance is conditional on a straightforward repair.

major comments (2)
  1. [§4, Theorem 4.5] The proof asserts 'Comme A est un B-module de présentation finie' after writing A = B[Z_1,...,Z_t]/a. This does not follow from the preceding definitions: A is a finitely presented B-algebra, not necessarily a finitely presented B-module, and a quotient of B[Z] by a finitely generated ideal can easily be non-finite over B. To conclude that C = S^{-1}A is a finite-dimensional L-vector space, one needs a Noether normalization making A finite over B = K[Y_1,...,Y_r]; the proof neither performs the change of variables that such a normalization requires nor cites a theorem ensuring its existence at this point. Without this step, the contradiction r>0 is not established, and the proof of Theorem 4.5, which is load-bearing for Theorem 4.4, is incomplete.
  2. [§4, Lemma 4.1] Lemma 4.1 is stated without proof, with only a remark that the stronger Lemma 4.2 from [1] exists and that the paper bases itself on Lemma 4.1. The text does not show how Lemma 4.1 follows from Lemma 4.2 and Lemma 1.3(5), even though such a derivation is short. Since Theorem 4.3, point 5 ⇒ 1, explicitly invokes Lemma 4.1, this missing derivation is a real gap, though it is easily fixed by adding the one-line argument.
minor comments (3)
  1. [Abstract] The French abstract contains the typo 'cors discret' for 'corps discret'.
  2. [§3, Lemma 3.3] The statement 'toute algèbre étale est une algèbre étale' uses the same word for the general étale of Definition 3.1(3) and the tracially etale of Definition 1.1; the underlining used elsewhere for the tracial notion is missing here and should be restored to avoid ambiguity.
  3. [§4, Theorem 4.5] The derivation ∂_1 : L → L is described as 'la dérivée partielle par rapport à X_1', but if X_1 is not among the variables Y_1,...,Y_r after the (implicit) relabeling, then ∂_1 is not a derivation on L; this needs clarification, ideally after the Noether normalization is made explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No construction-level circularity: the proof does not assume its target as an input; it has omitted proofs and self-citations, which are correctness gaps rather than circular reductions.

full rationale

The derivation chain is: netteness is defined as Omega_{A/K}=0 (Definition 3.1). Theorem 4.3 proves that a strictly finite nette algebra over a discrete field is tracially etale, using the unproved Lemma 4.1. Theorem 4.4 then reduces the remaining issue, proving that a nette algebra is strictly finite, to proving that its Noether dimension is at most 0, citing [ACMC, theoreme IV-8.16]. Finally Theorem 4.5 argues by contradiction that r>0 is incompatible with Omega=0. None of these steps sets the conclusion equal to a hypothesis by definition. The use of [ACMC, IV-8.16] is a self-citation, but it is a general structural equivalence, strictly finite iff Noether dimension at most 0, whose statement does not contain netteness or the target theorem, so it is independent support rather than a circular premise. The companion preprint [1] is cited for the stronger Lemma 4.2, but Lemma 4.2 is not actually used in the proof; the paper instead uses the weaker unproved Lemma 4.1. That is an omitted proof, not a circular reduction. A genuine gap is present in Theorem 4.5: the sentence "Comme A est un B-module de pr\'esentation finie" requires a Noether-normalization or change-of-variables theorem making A finite over B=K[Y_1,...,Y_r]; this is neither proved nor cited at that point, and "A = B[Z]/a" alone does not make C a finite-dimensional L-vector space. This is a correctness or completeness defect, as is the unproved Lemma 4.1, not a circularity: finite-as-a-B-module is not the same as the target conclusion finite-as-a-K-vector-space, and the missing theorem is external to the paper's own assumptions. Overall, the central claim is not forced by construction or by a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are involved since the paper is a pure mathematical proof. The main axiomatic input consists of standard constructive algebra results from [ACMC] and the unproved Lemma 4.1, which is load-bearing for Theorem 4.3.

assumptions (5)
  • domain assumption A discrete field has decidable equality and every element is zero or invertible.
    Used throughout the paper as the constructive setting; stated in the Introduction.
  • standard math Kaehler differential module Omega_{A/k} = Coker(Ja) and its exact sequences (Lemmas 2.5 and 2.6).
    Invoked in Theorem 4.5 to pass from Omega_{A/K}=0 to Omega_{C/K}=0 and to construct derivations.
  • standard math Primitive element theorem for etale algebras over an infinite field (Theorem 1.5).
    Used in Theorem 4.5 to write C = L[x] with a separable polynomial f.
  • standard math Noether dimension criterion: a finitely presented algebra over a discrete field is strictly finite iff its Noether dimension is at most 0 (Theoreme IV-8.16 of [ACMC]).
    Reduction step after Theorem 4.4, attributing the criterion to [ACMC].
  • ad hoc to paper Lemma 4.1: a nette strictly finite algebra over a discrete field is reduced.
    Stated without proof in Section 4; the surrounding text mentions Lemma 4.2 from [1] but does not derive 4.1.

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Pith. "Pith review of A decisive Theorem (Un th\'eor\`eme d\'ecisif)." pith.science (2026). https://pith.science/paper/OBYBPCGA

@misc{pith2026250607098,
  author       = {Pith},
  title        = {Pith review of: A decisive Theorem (Un th\'eor\`eme d\'ecisif)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBYBPCGA}},
  note         = {Machine review of arXiv:2506.07098}
}
read the original abstract

We give an elementary proof of the theorem which states that a finite unramified algebra over a discrete field is tracically \'etale. -- Nous donnons une d\'emonstration \'el\'ementaire du th\'eor\`eme selon lequel toute alg\`ebre nette sur un cors discret est \'etale, de dimension finie comme espace vectoriel et traciquement \'etale.

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Reference graph

Works this paper leans on

44 extracted references · 43 canonical work pages

  1. [1]

    Nous abr´ egeons⟨ ⟨traciquement ´ etale⟩ ⟩en ´ etale(soulign´ e)

    L’alg` ebreA est dite traciquement ´ etalesi le discriminant Disc A/k est inversible. Nous abr´ egeons⟨ ⟨traciquement ´ etale⟩ ⟩en ´ etale(soulign´ e). Consid´ erons maintenant le cas d’une alg` ebre strictement finie sur un corps discretL

    work page

  2. [2]

    Un ´ el´ ement deA est dit s´ eparable (surL) s’il annule un polynˆ ome s´ eparable

    work page

  3. [3]

    Lemme 1.2

    L’alg` ebreA est dite alg´ ebrique s´ eparable (surL) si tout ´ el´ ement deA est s´ eparable sur L. Lemme 1.2. Soit f un polynˆ ome unitaire deL[X] et L[x] = L[X]/⟨f ⟩ l’alg` ebre quotient

    work page

  4. [4]

    L[x] est ´ etalesi, et seulement si, f est s´ eparable

    work page

  5. [5]

    Lemme 1.3

    L[x] est r´ eduite si, et seulement si,f est sans facteur carr´ e. Lemme 1.3. Soit B ⊇ K une alg` ebre strictement finie

    work page

  6. [6]

    Si B est r´ eduite, pour touta ∈ B il existe un unique idempotent e ∈ K[a] tel que ⟨a⟩ = ⟨e⟩

    L’alg` ebreB est z´ ero-dimensionnelle1. Si B est r´ eduite, pour touta ∈ B il existe un unique idempotent e ∈ K[a] tel que ⟨a⟩ = ⟨e⟩. En outre, lorsque e = 1, c’est-` a-dire lorsque a est inversible, a−1 ∈ K[a]

    work page

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    (a) B est un corps discret

    Les propri´ et´ es suivantes sont ´ equivalentes. (a) B est un corps discret. (b) B est sans diviseur de z´ ero :xy = 0 ⇒ (x = 0 ou y = 0). (c) B est connexe 2 et r´ eduite. (d) Le polynˆ ome minimal surK de n’importe quel ´ el´ ement deB est irr´ eductible

    work page

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    En outre, B est ´ etalesur K si, et seulement si, elle est ´ etale sur L et L est ´ etalesur K

    Si K ⊆ L ⊆ B et si L est un corps discret strictement fini sur K, alors B est strictement finie sur L. En outre, B est ´ etalesur K si, et seulement si, elle est ´ etale sur L et L est ´ etalesur K

    work page

Show all 44 references
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    , er) est un syst` eme fondamental d’idempotents orthogonaux deB, B est ´ etalesur K si, et seulement si, chacune des composantes B[1/ei] est ´ etalesur K

    Si (e1, . . . , er) est un syst` eme fondamental d’idempotents orthogonaux deB, B est ´ etalesur K si, et seulement si, chacune des composantes B[1/ei] est ´ etalesur K

  2. [10]

    Si B est ´ etaleelle est r´ eduite

  3. [11]

    Th´ eor` eme 1.4(caract´ erisation desK-alg` ebres ´ etales)

    Si car(K) > [ B : K ] et si B est r´ eduite, elle est ´ etale. Th´ eor` eme 1.4(caract´ erisation desK-alg` ebres ´ etales). Soit B une K-alg` ebre strictement finie donn´ ee sous la formeK[x1, . . . , xn]. Les propri´ et´ es suivantes sont ´ equivalentes

  4. [12]

    Le polynˆ ome minimal surK de chacun des xi est s´ eparable

  5. [13]

    En particulier, un corps L qui est une extension galoisienne de K est ´ etalesur K

    B est alg´ ebrique s´ eparable surK. En particulier, un corps L qui est une extension galoisienne de K est ´ etalesur K. Th´ eor` eme 1.5(th´ eor` eme de l’´ el´ ement primitif). Soit B une K-alg` ebre ´ etale

  6. [14]

    Si K est infini ou si B est un corps discret, alors B est une alg` ebre monog` ene, pr´ ecis´ ement de la formeK[b] ≃ K[T ]/⟨f ⟩ pour un b ∈ B et un f ∈ K[T ] s´ eparable

  7. [15]

    B est un produit fini de K-alg` ebres ´ etalesmonog` enes

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    I.e., pour tout ´ el´ ementx, il existe n ∈ N et y ∈ B tels que xn(1 − yx) = 0

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    3 2 Le module des diff´ erentielles de Kh¨ aler Dans la suite de l’article, les k-alg` ebres sont toujours suppos´ ees de pr´ esentation finie

    I.e., tout idempotent est ´ egal ` a 0 ou 1. 3 2 Le module des diff´ erentielles de Kh¨ aler Dans la suite de l’article, les k-alg` ebres sont toujours suppos´ ees de pr´ esentation finie. D´ erivations D´ efinition et notation 2.1. Soit k un anneau commutatif, A une k-alg` eb...

  10. [18]

    On appelle k-d´ erivation deA dans M , une application k-lin´ eaireδ qui v´ erifie l’´ egalit´ e de Leibniz δ(ab) = aδ(b) + bδ(a)

  11. [19]

    On note Der k(A, M) le A-module des k-d´ erivations deA dans M

  12. [20]

    Lorsque le contexte est clair, Der( A) est une abr´ eviation pour Derk(A, A)

    Une d´ erivation deA ⟨ ⟨tout court ⟩ ⟩est une d´ erivation ` a valeurs dansA. Lorsque le contexte est clair, Der( A) est une abr´ eviation pour Derk(A, A). Note. Si δ ∈ Derk(A, M) et c ∈ A, l’application k-lin´ eairex 7→ cδ(x) est aussi une d´ erivation. Cela justifie le point...

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    On a δ(1) = 0 car 12 = 1, et donc δ|k = 0

  14. [22]

    On a δ(f (y)) = f ′(y)δ(y)

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    Le module des diff´ erentielles On consid` ere maintenant le cas d’une alg` ebre de pr´ esentation finie A = k[X1,

    On a δ(g(x)) = Pn i=1 ∂g ∂Xi (x)δ(xi). Le module des diff´ erentielles On consid` ere maintenant le cas d’une alg` ebre de pr´ esentation finie A = k[X1, . . . , Xn]/⟨f1, . . . , fs⟩ = k[x1, . . . , xn] = k[x]. On rappelle que la matrice jacobienne du syst` eme polynomial est ...

  16. [24]

    L’application d est une k-d´ erivation avecd(xi) = ei

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    Pour tout A-module M et toute k-d´ erivationδ : A → M , il existe une unique application A-lin´ eaireθ : Coker(Ja) → M telle que θ ◦ d = δ

    L’application d est une d´ erivation universelleau sens suivant. Pour tout A-module M et toute k-d´ erivationδ : A → M , il existe une unique application A-lin´ eaireθ : Coker(Ja) → M telle que θ ◦ d = δ. A d δ ''Coker(Ja) θ ! // M k-d´ erivations applications A-lin´ eaires. O...

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    Ceci permet de d´ efinir un homomorphisme canonique deAS-modules : φ : (Derk(A, M))S − →Derk(AS, MS)

    (a) Une k-d´ erivation deA dans M induit une unique d´ erivation deAS dans MS. Ceci permet de d´ efinir un homomorphisme canonique deAS-modules : φ : (Derk(A, M))S − →Derk(AS, MS). (b) L’homomorphisme φ est un isomorphisme si M ou ΩA/k est un A-module projectif de type fini. (...

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    Lemme 2.6 (extension d’une alg` ebre et d´ erivations)

    On a un isomorphisme naturel de AS-modules (ΩA/k)S → ΩAS /k. Lemme 2.6 (extension d’une alg` ebre et d´ erivations). Soient A une k-alg` ebre,A ρ − →B une alg` ebre etM un B-module 4

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    (b) Puisque M est naturellement un A-module, on a une application B-lin´ eaire canonique eρ : Derk(B, M) − →Derk(A, M) : δ 7→ δ ◦ ρ

    (a) Puisque toute A-d´ erivation deB dans M est aussi une k-d´ erivation on a une injection B-lin´ eaire canoniquej : DerA(B, M) → Derk(B, M). (b) Puisque M est naturellement un A-module, on a une application B-lin´ eaire canonique eρ : Derk(B, M) − →Derk(A, M) : δ 7→ δ ◦ ρ. O...

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    La suite d’applications B-lin´ eaires ci-dessous est exacte B ⊗k ΩA/k − →ΩB/k − →ΩB/A − →0 En particulier si ΩA/k = 0 alors ΩB/A = ΩB/k

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    Plus pr´ ecis´ ement, l’applicationA′-lin´ eaireA′ → k′ ⊗k ΩA/k est une d´ erivation universelle pour la k′-alg` ebreA′

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    5 3 Alg` ebres lisses, nettes, ´ etales D´ efinition 3.1.On consid` ere unek-alg` ebre de pr´ esentation finie A = k[X1,

    Malgr´ e le titre du lemmeρ n’est pas suppos´ e injectif. 5 3 Alg` ebres lisses, nettes, ´ etales D´ efinition 3.1.On consid` ere unek-alg` ebre de pr´ esentation finie A = k[X1, . . . , Xn]/⟨f1, . . . , fs⟩ et l’on note Ja la transpos´ ee de la matrice jacobienne

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    Ja est surjec- tive, ou encore DA,n(Ja) = ⟨1⟩

    La k-alg` ebreA est dite nette ou non ramifi´ eelorsque ΩA/k = 0, i.e. Ja est surjec- tive, ou encore DA,n(Ja) = ⟨1⟩

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    (b) La k-alg` ebreA est dite lisse s’il existe un syst` eme v1,

    (a) La k-alg` ebreA est dite pr´ esent´ ee comme lisse de base(standard smooth chez Stacks) si s ⩽ n et le premier mineur s × s de Ja est inversible dans A. (b) La k-alg` ebreA est dite lisse s’il existe un syst` eme v1, . . . , vr d’´ el´ ements co- maximaux de k telle que ch...

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    DA,n(Ja) = ⟨1⟩

    (a) La k-alg` ebreA est pr´ esent´ ee comme ´ etale de basesi s = n et det(Ja) ∈ A×, i.e. DA,n(Ja) = ⟨1⟩. On dit aussi dans ce cas que A est une alg` ebre de Newton. (b) La k-alg` ebreA est dite ´ etales’il existe un syst` emev1, . . . , vr d’´ el´ ements comaxi- maux de k tel...

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    lisse, ´ etale) surk, il en va de mˆ eme pourA′ sur k′

    Si A est nette (resp. lisse, ´ etale) surk, il en va de mˆ eme pourA′ sur k′

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    Si A′ est nette (resp

    Supposons que k′ est fid` element plate sur k. Si A′ est nette (resp. lisse, ´ etale) sur k′, il en va de mˆ eme pourA sur k. Principe local-global concret 3.5. Soit A une k-alg` ebre de pr´ esentation finie, et s1, . . . , sn ∈ k comaximaux. Notons ki = k[1/si] Alors A est ne...

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    6 4 Un th´ eor` eme d´ ecisif : alg` ebres nettes sur un corps discret Lemme 4.1

    Il s’agit de la k[1/vj]-alg` ebreA[1/ρ(vj)]. 6 4 Un th´ eor` eme d´ ecisif : alg` ebres nettes sur un corps discret Lemme 4.1. Soit K → A une alg` ebre nette et strictement finie sur un corps discret K. Alors A est r´ eduite. Le r´ esultat beaucoup plus fort suivant (lemme 4.2...

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    A est engendr´ ee par des ´ el´ ements s´ eparables surK

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    Tous les ´ el´ ements deA sont s´ eparables surK

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    A est isomorphe ` a un produit fini de K-alg` ebresK[X]/⟨gi⟩ pour des polynˆ omes unitaires s´ eparablesgi

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    A est nette. En outre, si K est infini ou si A est un corps, A est isomorphe ` a une alg` ebreK[x] = K[X]/⟨g⟩ o` ug est un polynˆ ome unitaire s´ eparable deK[X] (th´ eor` eme 1.5 de l’´ el´ ement primitif). Note. Dans le point 4, en posant g := g1 · · ·gr, si les gi sont deux...

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    Pour le d´ ecider il suffit de calculer le discriminant du polynˆ ome minimal dexi

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    On l’obtient en utilisant la machinerie locale-globale ´ el´ ementaire des anneaux z´ ero-dimensionnels r´ eduits d´ ecrite dans [ACMC, section IV-8]

    Le th´ eor` eme est ´ egalement valable dans le cas o` uK est seulement suppos´ e z´ ero-dimensionnel r´ eduit. On l’obtient en utilisant la machinerie locale-globale ´ el´ ementaire des anneaux z´ ero-dimensionnels r´ eduits d´ ecrite dans [ACMC, section IV-8]. 8 R ´EF ´ERENC...

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    (2025) Preprint

    Quitt´e C., Lombardi H. (2025) Preprint. Identit´ es alg´ ebriques permettant de d´ emontrer qu’une alg` ebre libre finie nette est traciquement ´ etale. https://arxiv. org/abs/2506.03851 6

    2025 arXiv

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