Arithmetic properties of blocks of consecutive integers

This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erd\H{o}s-Woods conjecture for each k>2.Comment: A slightly corrected and extended version of a paper which will appear in January 2017 in the book From Arithmetic to Zeta-functions published by Springe

A journey through Galois groups, irreducible polynomials and diophatine equations

Computing the Galois group of the splitting field of a given polynomial with integer coefficients over the rationals is a classical problem in modern algebra. A theorem of Van der Waerden [Wae] asserts that almost all (monic) polynomials in Z[x] have associated Galois group Sn, the symmetric group on n letters. Thus, cases where the associated Galois group is different from Sn are rare. Nevertheless, examples of polynomials where the associated Galois group is not Sn are well-known. For example, the Galois group of the splitting field of the polynomial xp − 1, p 3 prime, is cyclic of order p − 1. For the polynomial xp − 2, p > 3, the Galois group is the subgroup of Sp generated by a cycle of length p and a cycle of length p − 1. One interest in this paper is to find other collections of polynomials with integer or rational coefficients whose Galois groups are isomorphic to these groups.Natonal Science Foundation (Filaseta)Grants SEP-CONACyT 37259 E and 37260E (Luca

Galois Groups of Polynomials Arising from Circulant Matrices

Computng the Galois group of the splitting field of a given polynomial with interger coefficients is a classical problem in modern algebra. A theorem of Van de Waerden {Wae) asseerts that almost all (monic) polynomials in Z(x) have associated Galois group Sn, the symmetric on n letters. Thus, cases where the associated Galois group is different from Sn are rare. Nevertheless, examples of polynomials where the associated Galois group is no Sn are well-known..