Finite Galois Extension Research Articles

Classification of the types for which every Hopf–Galois correspondence is bijective

Let L/K be any finite Galois extension with Galois group G. It is known by Chase and Sweedler that the Hopf–Galois correspondence is injective for every Hopf–Galois structure on L/K, but it need not be bijective in general. Hopf–Galois structures are known to be related to skew braces, and recently, the first-named author and Trappeniers proposed a new version of this connection with the property that the intermediate fields of L/K in the image of the Hopf–Galois correspondence are in bijection with the left ideals of the associated skew brace. As an application, they classified the groups G for which the Hopf–Galois correspondence is bijective for every Hopf–Galois structure on any G-Galois extension. In this paper, using a similar approach, we shall classify the groups N for which the Hopf–Galois correspondence is bijective for every Hopf–Galois structure of type N on any Galois extension.

The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves

Abstract We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$ , and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F. The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.

The ring of finite algebraic numbers and its application to the law of decomposition of primes

In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field.

Profinite version of the Beckmann-Black problem

The Beckmann-Black problem asks whether any given finite Galois extension E/K of group G is the specialization at some point t 0 ∈ P 1 ( K ) of some finite regular Galois extension F / K ( T ) with the same group. In this paper, we study a generalization of this problem for infinite extensions, via the profinite twisting lemma, and apply the latter in many situations, like abelian infinite extensions and the l -universal Frattini cover of an arbitrary finite group.

The Ostrowski quotient of an elliptic curve

For [Formula: see text] a finite Galois extension of number fields, the relative Pólya group [Formula: see text] is the subgroup of the ideal class group of [Formula: see text] generated by all the strongly ambiguous ideal classes in [Formula: see text]. The notion of Ostrowski quotient [Formula: see text], as the cokernel of the capitulation map into [Formula: see text], has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define [Formula: see text] and [Formula: see text] which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For [Formula: see text] an elliptic curve defined over [Formula: see text], we define the Ostrowski quotient [Formula: see text] and the coarse Ostrowski quotient [Formula: see text] of [Formula: see text] relative to [Formula: see text], for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group [Formula: see text] and we analyze this theorem, in some details, for the class of curves [Formula: see text] over quadratic extensions [Formula: see text].

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.

Finite splittings of differential matrix algebras

It is well known that central simple algebras are split by suitable finite Galois extensions of their centers. In [4] a counterpart of this result was studied in the set up of differential matrix algebras, wherein Picard-Vessiot extensions that split matrix differential algebras were constructed. In this article, we exhibit instances of differential matrix algebras which are split by finite extensions. In some cases, we relate the existence of finite splitting extensions of a differential matrix algebra to the triviality of its tensor powers, and show in these cases, that orders of differential matrix algebras divide their degrees.

Triangulations of non-archimedean curves, semi-stable reduction, and ramification

Let K be a complete discretely valued field with algebraically closed residue field and let ℭ be a smooth projective and geometrically connected algebraic K-curve of genus g. Assume that g⩾2, so that there exists a minimal finite Galois extension L of K such that ℭ L admits a semi-stable model. In this paper, we study the extension L∣K in terms of the minimal triangulation of C, a distinguished finite subset of the Berkovich analytification C of ℭ. We prove that the least common multiple d of the multiplicities of the points of the minimal triangulation always divides the degree [L:K]. Moreover, in the special case when d is prime to the residue characteristic of K, then we show that d=[L:K], obtaining a new proof of a classical theorem of T. Saito. We then discuss curves with marked points, which allows us to prove analogous results in the case of elliptic curves, whose minimal triangulations we describe in full in the tame case. In the last section, we illustrate through several examples how our results explain the failure of the most natural extensions of Saito’s theorem to the wildly ramified case.

On matrix units of semisimple group algebras

We present two methods to explicitly compute complete sets of orthogonal primitive idempotents in simple components of semisimple group algebras over finite Galois extensions of the rationals and some classes of finite groups, including finite nilpotent groups.

Ostrowski quotients for finite extensions of number fields

For $L/K$ a finite Galois extension of number fields, the relative P\'olya group $\Po(L/K)$ coincides with the group of strongly ambiguous ideal classes in $L/K$. In this paper, using a well known exact sequence related to $\Po(L/K)$, in the works of Brumer-Rosen and Zantema, we find short proofs for some classical results in the literatur. Then we define the ``Ostrowski quotient'' $\Ost(L/K)$ as the cokernel of the capitulation map into $\Po(L/K)$, and generalize some known results for $\Po(L/\mathbb{Q})$ to $\Ost(L/K)$.

Arithmétique des extensions intérieures

In this article, we study the structure of finite inner Galois extensions and generalize several well-known properties of division algebras such as the Skolem-Noether theorem or the characterization to be a crossed product. We also describe, for any field K, the analoguous of the Brauer group for K: it is a certain subset of the Brauer group of the center of K, Br(Z(K)), which gives a one-to-one correspondance with the set of finite innner Galois and central extensions of K.

Equidistribution of αpθ$\alpha p^{\theta }$ with a Chebotarev condition and applications to extremal primes

We establish a joint distribution result concerning the fractional part of α p θ $\alpha p^\theta$ for θ ∈ ( 0 , 1 ) , α > 0 $\theta \in (0,1), \ \alpha >0$ , where p is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over Q $\mathbb {Q}$ . As an application, for a fixed non-CM elliptic curve E / Q $E/\mathbb {Q}$ , an asymptotic formula is given for the number of primes at the extremes of the Sato–Tate measure modulo a large prime ℓ. These are precisely the primes p for which the Frobenius trace a p ( E ) $a_p(E)$ satisfies the congruence a p ( E ) ≡ [ 2 p ] mod ℓ $a_p(E)\equiv [2\sqrt {p}] \bmod \ell$ . We assume a zero-free region hypothesis for Dedekind zeta functions of number fields.

On [formula omitted]-Hopf-Galois structures

Let K/F be a finite Galois extension of fields with Gal(K/F)=Γ. In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. The dihedral group is one particular example of a semidirect product of Zn and Z2. In this article we count the number of Hopf-Galois structures with Galois group Γ of type G, where Γ,G are groups of the form Zn⋊ϕZ2 when n is odd with radical of n being a Burnside number. As an application we also find the corresponding number of skew braces.

Theory and applications of linearized multivariate skew polynomials

In this work, linearized multivariate skew polynomials over division rings are introduced. Such polynomials are right linear over the corresponding centralizer and generalize linearized polynomial rings over finite fields, group rings or differential polynomial rings. Their natural evaluation is connected to the remainder-based evaluation of free multivariate skew polynomials. It is shown that P-independent sets are those given by right linearly independent sets when partitioned into conjugacy classes. Hence finitely generated P-closed sets correspond to lists of finite-dimensional right vector spaces, extending Lam and Leroy's results on univariate skew polynomials. It is also shown that products of free multivariate skew polynomials translate into coordinate-wise compositions of linearized multivariate skew polynomials, which in turn translate into matrix products over the corresponding centralizers. Later, linearized multivariate Vandermonde matrices are introduced, which generalize multivariate Vandermonde, Moore and Wronskian matrices. The previous results explicitly give their ranks in general. P-Galois extensions of division rings are then introduced, which generalize classical (finite) Galois extensions. Three Galois-theoretic results are generalized to such extensions: Artin's theorem on extension degrees, the Galois correspondence and Hilbert's Theorem 90.

On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Let E/K be a finite Galois extension of totally real number fields with Galois group G. Let p be an odd prime and let r>1 be an odd integer. The p-adic Beilinson conjecture relates the values at s=r of p-adic Artin L-functions attached to the irreducible characters of G to those of corresponding complex Artin L-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the ‘p-part’ of the equivariant Tamagawa number conjecture for the pair (h^0(mathrm {Spec}(E))(r), mathbb {Z}[G]). If r>1 is even we obtain a similar result for Galois CM-extensions after restriction to ‘minus parts’.

Formalizing Galois Theory

ABSTRACT We describe a project to formalize Galois theory using the Lean theorem prover, which is part of a larger effort to formalize all of the standard undergraduate mathematics curriculum in Lean. We discuss some of the challenges we faced and the decisions we made in the course of this project. The main theorems we formalized are the primitive element theorem, the fundamental theorem of Galois theory, and the equivalence of several characterizations of finite degree Galois extensions.

Multiplicities in Selmer groups and root numbers of Artin twists

Let K/F be a finite Galois extension of number fields and let σ be an absolutely irreducible, self-dual, complex valued representation of Gal(K/F). Let p be an odd prime and consider two elliptic curves E1,E2 defined over Q with good, ordinary reduction at primes above p and equivalent mod-p Galois representations. In this article, we study the variation of the parity of the multiplicities of σ in the representation space associated to the p∞-Selmer groups of E1 and E2 over K. We also compare the root numbers for the twists of E1 and E2 over F by σ and show that the p-parity conjecture holds for the twist of E1/F by σ if and only if it holds for the twist of E2/F by σ. We also express Mazur-Rubin-Nekovář's arithmetic local constants in terms of certain local Iwasawa invariants.

How far is an extension of p-adic fields from having a normal integral basis?

Let L/K be a finite Galois extension of p-adic fields with group G. It is well-known that OL contains a free OK[G]-submodule of finite index. We study the minimal index of such a free submodule, and determine it exactly in several cases, including for any cyclic extension of degree p of p-adic fields.

Equivariant Tamagawa number conjecture for Abelian varieties over global fields of positive characteristic

We state and prove certain cases of the equivariant Tamagawa number conjecture of a semistable Abelian variety over an everywhere unramified finite Galois extension of a global field of characteristic p > 0 p>0 under a semisimplicity hypothesis.

Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures

Let G G be a finite nonabelian group, and let ψ : G → G \psi :G\to G be a homomorphism with abelian image. We show how ψ \psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.