Related papers: On intersective polynomials with non-solvable Galo…

For a fixed prime power $q$ and natural number $d$ we consider a random polynomial $$f=x^n+a_{n-1}(t)x^{n-1}+\ldots+a_1(t)x+a_0(t)\in\mathbb F_q[t][x]$$ with $a_i$ drawn uniformly and independently at random from the set of all polynomials…

We present short elementary proofs of the well-known Ruffini-Abel-Galois theorems on insolvability of algebraic equations in radicals. These proofs are obtained from existing expositions by stripping away material not required for the…

For a differential operator $L$ of order $n$ over $C(z)$ with a finite (differential) Galois group $G\subset {\rm GL}(C^n)$, there is an algorithm, by M. van Hoeij and J.-A.~Weil, which computes the associated evaluation of the invariants…

In this paper we develop a theory of class invariants associated to $p$-adic representations of absolute Galois groups of number fields. Our main tool for doing this involves a new way of describing certain Selmer groups attached to…

Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a…

A result of Pyber states that every finite group $G$ contains an abelian subgroup whose order is quasi-polynomially large in $\lvert G\rvert$. We prove a similar result for $K$-approximate subgroups of solvable groups under only modest…

This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an…

We extend our method to compute division polynomials of Jacobians of curves over Q to curves over Q(t), in view of computing mod ell Galois representations occurring in the \'etale cohomology of surfaces over Q. Although the division…

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by…

We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a…

We enhance the analogy between field extensions and covering spaces by introducing the concept of splitting covering which correspondences to the splitting field in Galois theory. We define semi-topological Galois groups for Weierstrass…

Let F be a finitely generated field of characteristic zero and \Gamma<GL_n(F) a finitely generated subgroup. For an element g in \Gamma, let Gal(F(g)/ F) be the Galois group of the splitting field of the characteristic polynomial of g over…

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular , if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the…

The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2A2, 2B2 and 2G2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for $d$ the Coxeter…

In this paper we show some multiplicity estimates theorems for a connected algebraic group (not necessarily commutative) $G$ over an algebraically closed subfield of $\mathbb{C}$. More specifically, under particular assumptions on the…

For each pointed abelian group $(A,c)$, there is an associated {\em Galkin quandle} $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin…

We examine whether it is possible to realize finite groups $G$ as Galois groups of minimally tamely ramified extensions of $\mathbb{Q}$ and also specify both the inertia groups and the further decomposition of the ramified primes.

We present an algorithm to determine the Galois group of an irreducible monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree at most five. Following work of Conrad, Dummit, and Stauduhar this comes down to answering two questions: Is a given…

It was established before that fusion rings in a rational conformal field theory (RCFT) can be described as rings of polynomials, with integer coefficients, modulo some relations. We use the Galois group of these relations to obtain a local…

We determine all the $p$-adic analytic groups that are realizable as Galois groups of the maximal pro-$p$ extensions of number fields with prescribed ramification and splitting under an assumption which allows us to move away from the Tame…