Finite Separable Extension Research Articles

Skew bracoids

Skew braces are intensively studied owing to their wide ranging connections and applications. We generalize the definition of a skew brace to give a new algebraic object, which we term a skew bracoid. Our construction involves two groups interacting in a manner analogous to the compatibility condition found in the definition of a skew brace. We formulate tools for characterizing and classifying skew bracoids, and study substructures, quotients, homomorphisms, and isomorphisms. As a first application, we prove that finite skew bracoids correspond with Hopf-Galois structures on finite separable extensions of fields, generalizing the existing connection between finite skew braces and Hopf-Galois structures on finite Galois extensions.

Extendible functions and local root numbers: Remarks on a paper of R. P. Langlands

AbstractThis paper refers to Langlands' big set of notes devoted to the question if the (normalized) local Hecke–Tate root number , where E is a finite separable extension of a fixed nonarchimedean local field F, and χ a quasicharacter of , can be appropriately extended to a local ε‐factor for all virtual representations ρ of the corresponding Weil group . Whereas Deligne has given a relatively short proof by using the global Artin–Weil L‐functions, the proof of Langlands is purely local and splits into two parts: the algebraic part to find a minimal set of relations for the functions Δ, such that the existence (and unicity) of will follow from these relations; and the more extensive arithmetic part to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes, which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role.

Type-definable NIP fields are Artin–Schreier closed

Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p \gt 0$, then $K$ is Artin–Schreier closed (it has no Artin–Schreier extensions). As a consequence, $p$ does not divide the degree of any finite separable extension o

Arithmetic equivalence for non-geometric extensions of global function fields

In this paper we study couples of finite separable extensions of the function field Fq(T) which are arithmetically equivalent, i.e. such that prime ideals of Fq[T] decompose with the same inertia degrees in the two fields, up to finitely many exceptions. In the first part of this work, we extend previous results by Cornelissen, Kontogeorgis and Van der Zalm to the case of non-geometric extensions of Fq(T), which are fields such that their field of constants may be bigger than Fq. In the second part, we explicitly produce examples of non-geometric extensions of F2(T) which are equivalent and non-isomorphic over F2(T) and non-equivalent over F4(T), solving a particular Inverse Galois Problem.

Factoring differential operators over algebraic curves in positive characteristic

We present an algorithm for factoring linear differential operators with coefficients in a finite separable extension of F p ( x ). Our methods rely on specific tools arising in positive characteristic: p -curvature, structure of simple central algebras and p -Riccati equations.

Minimal Hopf-Galois Structures on Separable Field Extensions

In Hopf-Galois theory, every -Hopf-Galois structure on a field extension gives rise to an injective map from the set of -sub-Hopf algebras of into the intermediate fields of . Recent papers on the failure of the surjectivity of reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. In this paper we survey and illustrate group-theoretical methods to determine -Hopf-Galois structures on finite separable extensions in the extreme situation when has only two sub-Hopf algebras. This corresponds to the case when the lack of surjectivity is at its extreme.

The metabolicity index of involutions in characteristic two

We define the notion of metabolicity index of involutions of the first kind in characteristic two. It is shown that this index is preserved over odd degree extensions of the base field. Also, its behavior over finite separable extensions is studied. As an application, it is shown that an orthogonal involution on a central simple algebra of degree a power of two which is either anisotropic or metabolic is totally decomposable if it is totally decomposable over some separable extension of the ground field. This result is then used to strengthen an earlier result of the author which proves a characteristic two counterpart of a conjecture concerning Pfister involutions formulated by Bayer-Fluckiger et al.

À propos d’une version faible du problème inverse de Galois

This paper deals with the Weak Inverse Galois Problem, which, for a given field $k$, states that, for every finite group $G$, there exists a finite separable extension $L/k$ such that ${\rm {Aut}}(L/k)=G$. One of the paper’s goals is to explain how one ca

Ring extensions of length two

We characterize extensions of commutative rings [Formula: see text] such that [Formula: see text] is minimal for each [Formula: see text]-subalgebra [Formula: see text] of [Formula: see text] with [Formula: see text]. This property is equivalent to [Formula: see text] has length 2. Such extensions are either pointwise minimal or simple. We are able to compute the number of subextensions of [Formula: see text]. Besides commutative algebra considerations, our main result uses the concept of principal subfields of a finite separable field extension, which was recently introduced by van Hoeij et al. As a corollary of this paper, we get that simple extensions of length 2 have FIP.

On prolongations of valuations to the composite field

Let v be a Krull valuation of a field K with valuation ring Rv and K1,K2 be finite separable extensions of K which are linearly disjoint over K. Assume that the integral closure of Rv in the composite field K1K2 is a free Rv-module. For a given pair of prolongations v1,v2 of v to K1,K2 respectively, it is shown that there exists a unique prolongation w of v to K1K2 which extends both v1,v2. Moreover with Si as the integral closure of Rv in Ki, if the ring S1S2 is integrally closed and the residue field of v is perfect, then f(w/v)=f(v1/v)f(v2/v), where f(v′/v) stands for the degree of the residue field of a prolongation v′ of v over the residue field of v. As an application, it is deduced that if K1,K2 are algebraic number fields which are linearly disjoint over K=K1∩K2, then the number of prime ideals of the ring AK1K2 of algebraic integers of K1K2 lying over a given prime ideal ℘ of AK equals the product of the numbers of prime ideals of AKi lying over ℘ for i=1,2.

Isomorphism problems for Hopf–Galois structures on separable field extensions

Let L/K be a finite separable extension of fields whose Galois closure E/K has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf–Galois structure on L/K has the form E[N]G for some group N of order [L:K]. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf–Galois structures on a cyclic extension of degree pn, for p an odd prime number.

Jumps and Motivic Invariants of Semiabelian Jacobians

Abstract We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called jumps. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud’s description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

On the compositum of integral closures of valuation rings

It is well known that if K1,K2 are algebraic number fields with coprime discriminants, then the composite ring AK1AK2 is integrally closed and K1,K2 are linearly disjoint over the field of rationals, AKi being the ring of algebraic integers of Ki. In an attempt to prove the converse of the above result, in this paper we prove that if K1,K2 are finite separable extensions of a valued field (K,v) of arbitrary rank which are linearly disjoint over K=K1∩K2 and if the integral closure Si of the valuation ring Rv of v in Ki is a free Rv-module for i=1,2 with S1S2 integrally closed, then the discriminant of either S1/Rv or of S2/Rv is the unit ideal. We quickly deduce from this result that for algebraic number fields K1,K2 linearly disjoint over K=K1∩K2 for which AK1AK2 is integrally closed, the relative discriminants of K1/K and K2/K must be coprime.

New points on curves

Let K be a field and let L/K be a finite extension. Let X/K be a scheme of finite type. A point of X(L) is said to be new if it does not belong to the union of X(F), when F runs over all proper subextensions of L. Fix now an integer g>0 and a finite separable extension L/K of degree d. We investigate in this article whether there exists a smooth proper geometrically connected curve of genus g with a new point in X(L). We show for instance that if K is infinite of characteristic different from 2 and g is bigger or equal to [d/4], then there exist infinitely many hyperelliptic curves X/K of genus g, pairwise non-isomorphic over the algebraic closure of K, and with a new point in X(L). When d is between 1 and 10, we show that there exist infinitely many elliptic curves X/K with pairwise distinct j-invariants and with a new point in X(L).

Discriminant as a product of local discriminants

Let [Formula: see text] be a discrete valuation ring with maximal ideal [Formula: see text] and [Formula: see text] be the integral closure of [Formula: see text] in a finite separable extension [Formula: see text] of [Formula: see text]. For a maximal ideal [Formula: see text] of [Formula: see text], let [Formula: see text] denote respectively the valuation rings of the completions of [Formula: see text] with respect to [Formula: see text]. The discriminant satisfies a basic equality which says that [Formula: see text]. In this paper, we extend the above equality on replacing [Formula: see text] by the valuation ring of a Krull valuation of arbitrary rank and completion by henselization. In the course of proof, we prove a generalization of the well-known weak Approximation Theorem which is of independent interest as well.

The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation

Suppose that $R$ is a 2 dimensional excellent local domain with quotient field $K$, $K^*$ is a finite separable extension of $K$ and $S$ is a 2 dimensional local domain with quotient field $K^*$ such that $S$ dominates $R$. Suppose that $\nu^*$ is a valuation of $K^*$ such that $\nu^*$ dominates $S$. Let $\nu$ be the restriction of $\nu^*$ to $K$. The associated graded ring ${\rm gr}_{\nu}(R)$ was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension $(K,\nu)\rightarrow (K^*,\nu^*)$ of valued fields is without defect if and only if there exist regular local rings $R_1$ and $S_1$ such that $R_1$ is a local ring of a blow up of $R$, $S_1$ is a local ring of a blowup of $S$, $\nu^*$ dominates $S_1$, $S_1$ dominates $R_1$ and the associated graded ring ${\rm gr}_{\nu^*}(S_1)$ is a finitely generated ${\rm gr}_{\nu}(R_1)$-algebra. We also investigate the role of splitting of the valuation $\nu$ in $K^*$ in finite generation of the extensions of associated graded rings along the valuation. We will say that $\nu$ does not split in $S$ if $\nu^*$ is the unique extension of $\nu$ to $K^*$ which dominates $S$. We show that if $R$ and $S$ are regular local rings, $\nu^*$ has rational rank 1 and is not discrete and ${\rm gr}_{\nu^*}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra, then $\nu$ does not split in $S$. We give examples showing that such a strong statement is not true when $\nu$ does not satisfy these assumptions. We deduce that if $\nu$ has rational rank 1 and is not discrete and if $R\rightarrow R'$ is a nontrivial sequence of quadratic transforms along $\nu$, then ${\rm gr}_{\nu}(R')$ is not a finitely generated ${\rm gr}_{\nu}(R)$-algebra.

Commuting Hopf–Galois structures on a separable extension

ABSTRACTLet L∕K be a finite separable extension of local or global fields in any characteristic, let be two Hopf algebras giving Hopf–Galois structures on the extension, and suppose that the actions of on L commute. We show that a fractional ideal 𝔅 of L is free over its associated order in H1 if and only if it is free over its associated order in H2. We also study which properties these associated orders share.

On stabilizers of algebraic function fields of one variable

AbstractLetK~$\tilde K$be a fixed algebraic closure of an infinite fieldK. We consider an absolutely integral curveΓinPKn$\mathbb{P}_{K}^{n}$withn≥ 2. The curveΓK~$\it\Gamma_{\tilde{K}}$should have only finitely many inflection points, finitely many double tangents, and there exists no point inPK~n$\mathbb{P}_{\tilde{K}}^{n}$through which infinitely many tangents toΓK~$\it\Gamma_{\tilde{K}}$go. In addition there exists a prime numberqsuch thatΓK~$\it\Gamma_{\tilde{K}}$has a cusp of multiplicityqand the multiplicities of all other points ofΓK~$\it\Gamma_{\tilde{K}}$are at mostq. Under these assumptions, we construct a non-empty Zariski-open subsetOofPK~n$\mathbb{P}_{\tilde{K}}^{n}$such that ifn≥ 3, the projection from each pointo∈O(K) birationally mapsΓonto an absolutely integral curveΓ′inPKn−1$\mathbb{P}_{K}^{n-1}$with the same properties asΓ(keepingqunchanged). Ifn= 2, then the projection from eacho∈O(K) mapsΓontoPK1$\mathbb{P}_{K}^{1}$and leads to astabilizingelementtof the function fieldFofΓoverK. The latter means thatF/K(t) is a finite separable extension whose Galois closureF^${\hat F}$is regular overK.

Genus fields of congruence function fields

Let k be a rational congruence function field and consider a finite separable extension K/k. We consider the extension K/k satisfying the following condition. For each prime in k at least one prime in K above it is tamely ramified. Then, except for constants, we find the genus field Kge of K/k. In general, we describe the genus field of a global function field. We present some applications and examples of the main results.

Minimal fields of definition for Galois action

Let K be a field, and let f:X→Y be a finite étale cover between reduced and geometrically irreducible K-schemes of finite type such that fKs is Galois. Assuming f admits a Galois K-form f¯:X¯→Y, we use it to analyze fields of definition over K for the Galois property of f and the presence of K-points in general K-forms f′:X′→Y over Y(K).Additionally, we show that if K is Hilbertian, the group G is non-abelian, and the base variety is rational, then there are finite separable extensions L/K such that some L-form of fL does not descend to a cover of Y.