### Short interval results for k-free values of irreducible polynomials

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

This paper deals with the irreducibility of the mth order Bernoulli polynomials of degree m. As m tends to infinity, Eisenstein's criterion is shown to imply irreducibility for asymptotically > 1/5 of these polynomials.

We answer three reducibility (or irreducibility) questions for $0,1$-polynomials, those polynomials which have every coefficient either $0$ or $1$. The first concerns whether a naturally occurring sequence of reducible polynomials is finite. The second is whether every nonempty finite subset of an infinite set of positive integers can be the set of positive exponents of a reducible $0,1$-polynomial. The third is the analogous question for exponents of irreducible $0,1$-polynomials.

If f(x) and g(x) are relatively prime polynomials in ℤ[x] satisfying certain conditions arising from a theorem of Capelli and if n is an integer > N for some sufficiently large N, then the non-reciprocal part of f(x)xⁿ + g(x) is either identically ±1 or is irreducible over the rationals. This result follows from work of Schinzel in 1965. We show here that under the conditions that f(x) and g(x) are relatively prime 0,1-polynomials (so each coefficient is either 0 or 1) and f(0) = g(0) = 1, one...

We show that the discriminant of the generalized Laguerre polynomial ${L}_{n}^{\left(\alpha \right)}\left(x\right)$ is a non-zero square for some integer pair $(n,\alpha )$, with $n\ge 1$, if and only if $(n,\alpha )$ belongs to one of $30$ explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of ${L}_{n}^{\left(\alpha \right)}\left(x\right)$ over $\mathbb{Q}$ is the alternating group ${A}_{n}$. For example, we establish that for all but finitely many positive integers $n\equiv 2\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}4)$, the only $\alpha $ for which the Galois group of ${L}_{n}^{\left(\alpha \right)}\left(x\right)$ over $\mathbb{Q}$ is ${A}_{n}$ is $\alpha =n$.

**Page 1**