Irreducibility of Certain Quadrinomials

On the Reducibility of Certain Quadrinomials

In 2007 West Coast Number Theory conference Walsh asked to determine all irreducible polynomials of the form P (x) = x + x + x + 4 with integer exponents i > j > k > 0, such that for some positive integer l the polynomial P (x) is reducible in Z[x]. In this paper we prove that such polynomials are quadrinomials x4m + x3m + x2m + 4, where m is an odd positive integer. In addition, Walsh asked fo...

Irreducibility of certain pseudovarieties 1

We prove that the pseudovarieties of all finite semigroups, and of all aperiodic fin1 e sernigroups are irreducible for join, for semidirect product and for hlal 'ce~. product. In particular, these pseudovarieties do not admit maximal proper suhpseudovariet~es. More generally, analogous results are proved for the pseudovar~ety of all finite serrigroups all of whose subgroups are in a fixed pseu...

The Irreducibility of Certain Pure-cycle Hurwitz Spaces

We study “pure-cycle” Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-0 covers, using a combination of limit linear series theory and group theory to show that these spaces are always irreducible. In the case of four branch points, we also compute the associated Hurwitz numbers. Finally, we give a...

On the Irreducibility of a Certain Class of Laguerre Polynomials

Two Diophantine Approaches to the Irreducibility of Certain Trinomials

In the previous paper [FLSU], we determined the Galois groups associated with some polynomials constructed through the use of circulant matrices. In the process of determining the Galois groups, the irreducibility of the trinomial x + x +m was established, where p represents an odd prime and m an integer ≥ 2. The approach there was based on a method of Lebesgue [Le]. In this paper, we discuss t...