Some Galois groups over number fields

Some conditions are stated which imply that certain finite groups are Galois groups over some number fields and related fields.     

Frobenius Galois groups over quadratic fields

There exists a quadratic fieldQ(√D) over which every Frobenius group is realizable as a Galois group.     

Computation of Galois groups over function fields

Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.

     

Galois cohomology of the classical groups over imperfect fields

Elaborating on techniques of Bayer-Fluckiger and Parimala, we prove the following strong version of Serre’s Conjecture II for classical groups: let G be a simply connected absolutely simple group of outer type A_n or of type B_n, C_n or D_n (non trialitarian) defined over an arbitrary field F. If the separable dimension of F is at most 2 for every torsion prime of G, then every G-torsor is trivial.     

Period-index and u-invariant questions for function fields over complete discretely valued fields

Let K be a complete discretely valued field with residue field κ and F the function field of a curve over K. Let p be the characteristic of κ and ? a prime not equal to p. If the Brauer ?-dimensions of all finite extensions of κ are bounded by d and the Brauer ?-dimensions of all extensions of κ of transcendence degree at most 1 are bounded by d+1, then it is known that the Brauer ?-dimension of F is at most d+2 (Lieblich in J. Reine Angew. Math. 659:1–41, 2011; Saltman in J. Ramanujan Math. Soc. 12:25–47, 1997; Harbater et al. in Invent. Math. 178:231–263, 2009). In this paper we give a bound for the Brauer p-dimension of F in terms of the p-rank of κ. As an application, we show that if κ is a perfect field of characteristic 2, then any quadratic form over F in at least 9 variables is isotropic. This leads to the fact that every element in \(H^{3}(F,\mathbb{Z}/2\mathbb{Z})\) is a symbol. If κ is a finite field of characteristic 2, u(F)=8 is a result of Heath-Brown/Leep (Heath-Brown in Compos. Math. 146:271–287, 2010; Leep in J. Reine Angew. Math., 2013, to appear).     

Galois groups and complete domains

Consider a domain that is complete with respect to a non-zero prime ideal. This paper proves two Galois-theoretic results about such rings. Using Grothendieck’s Existence Theorem we prove that every finite group occurs as the Galois group of a Galois extension of . This generalizes results of David Harbater who proved the result in the case where the ideal is maximal and the domain is normal. As a consequence, we deduce that if is a Noetherian domain that is complete with respect to a non-zero prime ideal, then every finite group occurs as a Galois group over . This proves the Noetherian case of a conjecture posed by Moshe Jarden.     

Galois stratification over Frobenius fields

New first-order conformally covariant differential operators P_k on spinor-k-forms, i.e., tensor products of contravariant spinors with k-forms, in an arbitrary n-dimensional pseudo-Riemannian spin manifold, are introduced. This provides a series of generalizations of the Dirac operator $\text{?}\text{?}$, in analogy with the series of generalizations (introduced by the author in [1]) of the Maxwell operator and the conformally covariant Laplacian $\text{□}$ on functions. In particular, new intertwining operators for representations of SU(2, 2) and SO(p + 1, q + 1) are found. Related nonlinear covariant operators are also introduced, and mixed nonlinear covariant systems are obtained by coupling to the Yang-Mills-Higgs-Dirac system in dimension 4. The spinor-form bundle is isomorphic with E⁽³⁾ = E ? E ? E, where E is the spin bundle, and the P_k give a covariant operator on sections of E⁽³⁾. This is generalized to a covariant operator on E^{(2l + 1)}. The relation of powers of these operators to higher-order covariant operators on lower spin bundles (analogous to the relation between $\text{?}\text{?}$ and $\text{□}$) is discussed.     

On the u-invariant of function fields of curves over complete discretely valued fields

Let K be a complete discretely valued field with residue field κ  . If char(K)=0$\mathrm{char}(K)=0$, char(κ)=2$\mathrm{char}(\kappa )=2$ and [κ:κ²]=d$[\kappa :{\kappa }^2]=d$, we prove that there exists an integer N depending on d such that the u-invariant of any function field in one variable over K is bounded by N. The method of proof is via introducing the notion of uniform boundedness for the p-torsion of the Brauer group of a field and relating the uniform boundedness of the 2-torsion of the Brauer group to the finiteness of the u-invariant. We prove that the 2-torsion of the Brauer group of function fields in one variable over K is uniformly bounded.     

Galois theory over complete local domains

Complete local domains play an important role in commutative algebra and algebraic geometry, and their algebraic properties were already described by Cohen’s structure theorem in 1946. However, the Galois theoretic properties of their quotient fields only recently began to unfold. In 2005 Harbater and Stevenson considered the two dimensional case. They proved that the absolute Galois group of the field K((X, Y)) (where K is an arbitrary field) is semi-free. In this work we settle the general case, and prove that if R is a complete local domain of dimension exceeding 1, then the quotient field of R has a semi-free absolute Galois group.     

Abstract simplicity of complete Kac-Moody groups over finite fields

Let G be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix A, as constructed by Tits. It is known that G admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac-Moody group which is defined to be the closure of G in the automorphism group of its building. Our main goal is to determine when complete Kac-Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices A of rank at least four. Our proof uses Tits’ simplicity theorem for groups with a BN-pair and methods from the theory of pro-p groups.     

Galois LCD codes over finite fields

In this paper, we generalize the linear complementary dual codes (LCD codes for short) to k-Galois LCD codes, and study them by a uniform method. A necessary and sufficient condition for linear codes to be k-Galois LCD codes is obtained, two classes of k-Galois LCD MDS codes are exhibited. Then, necessary and sufficient conditions for λ-constacyclic codes being k-Galois LCD codes are characterized. Some classes of k-Galois LCD λ-constacyclic MDS codes are constructed. Finally, we study Hermitian LCD λ-constacyclic codes, and present a class of Hermitian LCD λ-constacyclic MDS codes.     

Galois cohomology of the classical groups over fields of cohomological dimension≦2

Oblatum 8-XII-1994 7-III-1995     

Galois groups of intersections of local fields

LetK be a denumerable Hilbertian field with separable algebraic closure and Galois group , letw ₁,...w _n be absolute values on . Then for almost allσ ∈ G _K ⁿ (in the sense of Haar measure) there are no relations between the decomposition groups G _K (ω ₁ σ ₁),...,G _K (w _n σ _n ) of the absolute valuesw ₁ σ ₁,...,w _n σ _n i.e. the subgroup of G _K generated by these groups is the free product of these groups.     

Integer valued polynomials over function fields

Let A be an integrally closed subring of a function field K defined over a finite field. In this paper we investigate whether the subring of K[X], consisting of those polynomials ƒ with ƒ[A]⊂A, has an A-basis {g_i: i ∈ ℤZ_≥0}, with deg (g_i) = i.     

Computing Galois groups over the rationals

Practical computational techniques are described to determine the Galois group of a polynomial over the rationals, and each transitive permutation group of degree 3 to 7 is realised as a Galois group over the rationals. The exact computations furnish a proof of the result.     

Relative Brauer groups of discrete valued fields

Let be a non-trivial finite Galois extension of a field . In this paper we investigate the role that valuation-theoretic properties of play in determining the non-triviality of the relative Brauer group, , of over . In particular, we show that when is finitely generated of transcendence degree 1 over a -adic field and is a prime dividing , then the following conditions are equivalent: (i) the -primary component, , is non-trivial, (ii) is infinite, and (iii) there exists a valuation of trivial on such that divides the order of the decomposition group of at .

     

Spherically complete models of Hensel minimal valued fields

We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is 0-h^mix-minimality (which, in equi-characteristic 0, amounts to 0-h-minimality).