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.     

On differential Galois groups of strongly normal extensions

We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p‐valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in , we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorphisms.     

Galois groups of some iterated polynomials over cyclotomic extensions

Let \(\varphi _p(z)=(z-1)^p+2-\zeta _p\), where \(\zeta _p\in \bar{\mathbb {Q}}\) is a primitive pth root of unity. Building on previous work, we show that the nth iterate \(\varphi _p^n(z)\) has Galois group \([C_p]^n\), an iterated wreath product of cyclic groups, whenever p is not a Wieferich prime.     

On normal radical extensions of the rationals

Lower bounds on the bandwidth, the size of a vertex separator of general undirected graphs, and the largest common subgraph of two undirected (weighted) graphs are obtained. The bounds are based on a projection technique developed for the quadratic assignment problem, and once more demonstrate the importance of the extreme eigenvalues of the Laplacian. They will be shown to be strict for certain classes of graphs and compare favourably to bounds already known in literature. further improvement is gained by applying nonlinear optimization methods.     

On Galois groups of unramified pro-p extensions

Let p be an odd prime satisfying Vandiver’s conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z _p -extensions of Q(μ _p ) and the Galois group of the maximal unramified pro-p extension of Q . We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for to be abelian. The bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p-units. We obtain in particular that, for p  <  1,000, Greenberg’s conjecture that X is pseudo-null holds and is in fact abelian.     

On the embedding of central extensions into permutation wreath products

We find a necessary condition for embedding a central extension of a group G with elementary abelian kernel into the wreath product that corresponds to a given permutation action of G.     

On the isomorphism of wreath products of groups

The well-known Neumann theorem on the isomorphism of standard wreath products is generalized to the wreath products of an arbitrary transitive permutation group and an abstract group. Pedagogical Institute, Vinnitsa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 671–679, June, 1994     

Galois extensions as functors of comodules

Let A be a finite Hopf algebra over a commutative ring k. We show a one-to-one correspondence between the A-Galois extensions of k and certain functors from the category of A-comodules to the category of k-modules.     

The differential Galois theory of strongly normal extensions

Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin's elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a certain tensor product.

We identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product, thus identifying the Galois group with the closed points of an affine differential scheme. Moreover, the tensor product has a natural coring structure which translates into the Galois group operation: composition of automorphisms.

This affine differential scheme splits, i.e. is obtained by base extension from a (not differential, not necessarily affine) group scheme. As a consequence, the Galois group is canonically isomorphic to the closed, or rational, points of a group scheme defined over constants. We obtain the fundamental theorem of differential Galois theory, giving a bijective correspondence between subgroup schemes and intermediate differential fields.

On the way to this result we study certain aspects of differential algebraic geometry, e.g. closed immersions, products, local ringed space of constants, and split differential schemes.

     

Subgroup distortion in wreath products of cyclic groups

We study the effects of subgroup distortion in the wreath products , where A is finitely generated abelian. We show that every finitely generated subgroup of has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of having distortion function equivalent to the given polynomial. Also, a formula for the length of elements in arbitrary wreath product easily shows that the group has distorted subgroups, while the lamplighter group has no distorted (finitely generated) subgroups. In the course of the proof, we introduce a notion of distortion for polynomials. We are able to compute the distortion of any polynomial in one variable over Z,R or C.     

Representation theory of wreath products of finite groups

This is an introduction to the representation theory of wreath products of finite groups. We also discuss in full details a couple of examples. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

Twisted burnside theory for the discrete Heisenberg group and for wreath products of some groups

For each positive integer N, an automorphism with the Reidemeister number 2N of the discrete Heisenberg group is constructed; an example of determination of points in the unitary dual object being fixed with respect to the mapping induced by the group automorphism is given. For wreath products of finitely generated Abelian groups and the group of integers, it is proved that if the Reidemeister number of an arbitrary automorphism is finite, then it is equal to the number of fixed points of the induced mapping on a finite-dimensional part of the unitary dual object.     

On semidirect products and wreath products of partially ordered groups

The notions of Cartesian and semidirect products for partially ordered groups are considered. A series of results on those products of AO mathcal{A}mathcal{O} -groups and interpolation groups is obtained. Some results concerning wreath products of directed groups are obtained.     

Galois groups of some vectorial polynomials

Previously nice vectorial equations were constructed having various finite classical groups as Galois groups. Here such equations are constructed for the remaining classical groups. The previous equations were genus zero equations. The present equations are strong genus zero.