Galois 2-extensions unramified outside 2

We classify quadratic, biquadratic and degree 4 cyclic 2-rational number fields. We also classify those quadratic number fields which are not 2-rational, but have a degree 2 extension, which is Galois over Q and is 2-rational. In this case we explicitly describe the Galois group of their maximal pro-2 extension unramified outside 2 and infinity using a result of Herfort-Ribes-Zalesskii on virtually free pro-p groups.     

Polynomials with PSL(2, 7) as Galois group

We describe a technique for determining the set-transitivity of the Galois group of a polynomial over the rationals. As an application we give a short proof that the polynomial P₇(x) = x⁷ ? 154x + 99 has the simple group PSL(2, 7) of order 168 as its Galois group over the rationals. A similar method is used to prove that the associated splitting field is not that of the polynomial x⁷ ? 7x + 3 given by Trinks [9].     

Groups of p-absolute Galois type that are not absolute Galois groups

Let p be a prime. We study pro-p groups of p-absolute Galois type, as defined by Lam–Liu–Sharifi–Wake–Wang. We prove that the pro-p completion of the right-angled Artin group associated to a chordal simplicial graph is of p-absolute Galois type, and moreover it satisfies a strong version of the Massey vanishing property. Also, we prove that Demushkin groups are of p-absolute Galois type, and that the free pro-p product — and, under certain conditions, the direct product — of two pro-p groups of p-absolute Galois type satisfying the Massey vanishing property, is again a pro-p group of p-absolute Galois type satisfying the Massey vanishing property. Consequently, there is a plethora of pro-p groups of p-absolute Galois type satisfying the Massey vanishing property that do not occur as absolute Galois groups.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

The Merkuriev-Suslin theorem for any semi-local ring

We introduce here a method which uses étale neighborhoods to extend results from smooth semi-local rings to arbitrary semi-local rings A by passing to the henselization of a smooth presentation of A. The technique is used to show that étale cohomology of A agrees with Galois cohomology, to show that the Merkuriev-Suslin theorem holds for A, and to describe torsion in K₂(A).     

Rational curve with Galois point and extendable Galois automorphism

Let G be the Galois group of a Galois point for a plane curve C. An element of G induces a birational transformation of C. We study if it can be extended to a projective or birational transformation of the plane. In the course of the study we give the defining equation of a rational curve with the Galois point. Furthermore, we introduce a special birational transformation in order to make the defining equation into a simpler form.     

On exceptional pq-groups

A finite group G is called exceptional if for a Galois extension F/k of number fields with the Galois group G,in the Brauer-Kuroda relation of the Dedekind zeta functions of fields between k and F,the zeta function of F does not appear.In the present paper we describe effectively all exceptional groups of orders divisible by exactly two prime numbers p and q,which have unique subgroups of orders p and q.     

Un D-groupoïde de Galois local pour les systèmes aux q-différences fuchsiens

We compute the Galois D-groupoid of a constant linear q-difference system and we define, giving its realisations, a local Galois D-groupoid for fuchsian q-difference systems.     

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel n-Unipotent Conjecture and the Vanishing n  -Massey Conjecture for n≥3$n\ge 3$ were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of p-maximal extensions are free pro-p-groups. We also prove the Kernel Unipotent Conjecture for Demushkin groups of rank 2, and establish various filtration results for free pro-p-groups, provide examples of pro-p-groups which do not have the kernel n-unipotent property, compare various Zassenhaus filtrations with the descending p-central series and establish new type of automatic Galois realization.     

Artin t-motifs

We show that analytically trivial t-motifs satisfy a Tannakian duality, without restrictions on the base field, save for that it be of generic characteristic. We show that the group of components of the t-motivic Galois group coincides with the absolute Galois group of the base field.     

The Weierstrass semigroup of a pair of GaloisWeierstrass points with prime degree on a curve

We describe the Weierstrass semigroup of a Galois Weierstrass point with prime degree and the Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree, where a Galois Weierstrass point with degree n means a total ramification point of a cyclic covering of the projective line of degree n.*Supported by Korea Research Foundation Grant (KRF-2003-041-C20010).**Partially supported by Grant-in-Aid for Scientific Research (15540051), JSPS.     

Noncommutative Galois Extension and Graded <Emphasis Type="Italic">q</Emphasis>-Differential Algebra

We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of semi-commutative Galois extension induced by the graded q-differential algebra. As an example we consider the quaternions which can be viewed as the semi-commutative Galois extension of complex numbers.     

ℒ-invariants and logarithm derivatives of eigenvalues of Frobenius

Let K be a p-adic local field. We study a special kind of p-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms. In particular, we verify that a formula of Colmez can be generalized to our case. We also include a degenerated version of Colmez’s formula.     

Motivic cohomology of the simplicial motive of a Rost variety

We compute the motivic cohomology groups of the simplicial motive X_θ of a Rost variety for an n-symbol θ in Galois cohomology of a field. As an application we compute the kernel and cokernel of multiplication by θ in Galois cohomology. We also show that the reduced norm map on K₂ of a division algebra of square-free degree is injective.     

G-Closed fields and imbeddings of quadratic number fields

A field, K, that has no extensions with Galois group isomorphic to G is called G-closed. It is proved that a finite extension of K admits an infinite number of nonisomorphic extensions with Galois group G. A trinomial of degree n is exhibited with Galois group, the symmetric group of degree n, and with prescribed discriminant. This result is used to show that any quadratic extension of an A_n-closed field admits an extension with Galois group A_n.     

Commutative Abelian Galois extensions of a Banach algebra

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C(X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.     

On local Galois representations associated to ordinary Hilbert modular forms

Let F be a totally real field and p be an odd prime which splits completely in F. We show that a generic p-ordinary non-CM primitive Hilbert modular cuspidal eigenform over F of parallel weight two or more must have a locally non-split p-adic Galois representation, at at least one of the primes of F lying above p. This is proved under some technical assumptions on the global residual Galois representation. We also indicate how to extend our results to nearly ordinary families and forms of non-parallel weight.     

Equivariant Brauer and Picard groups and a Chase–Harrison–Rosenberg exact sequence

The classical Chase–Harrison–Rosenberg exact sequence relates the Picard and Brauer groups of a Galois extension S of a commutative ring R to the group cohomology of the Galois group. We associate to each action of a locally compact group G on a locally compact space X two groups which we call the equivariant Picard group and the equivariant Brauer group. We then prove an analogue of the Chase–Harrison–Rosenberg exact sequence in the which the roles of the Picard and Brauer groups are played by their equivariant analogues.     

Higher Hopf formulae for homology via Galois Theory

We use Janelidze's Categorical Galois Theory to extend Brown and Ellis's higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A is the category Gp of all groups and B is the category Ab of all abelian groups, this yields a new proof for Brown and Ellis's formulae. We also give explicit formulae in the cases of groups vs. k-nilpotent groups, groups vs. k-solvable groups and precrossed modules vs. crossed modules.