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Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

### On mth order Bernoulli polynomials of degree m that are Eisenstein

Colloquium Mathematicae

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.

### On three questions concerning $0,1$-polynomials

Journal de Théorie des Nombres de Bordeaux

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.

### On the irreducibility of 0,1-polynomials of the form f(x)xⁿ + g(x)

Colloquium Mathematicae

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...

### On classifying Laguerre polynomials which have Galois group the alternating group

Journal de Théorie des Nombres de Bordeaux

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 $\left(n,\alpha \right)$, with $n\ge 1$, if and only if $\left(n,\alpha \right)$ 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 $ℚ$ is the alternating group ${A}_{n}$. For example, we establish that for all but finitely many positive integers $n\equiv 2\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right)$, the only $\alpha$ for which the Galois group of ${L}_{n}^{\left(\alpha \right)}\left(x\right)$ over $ℚ$ is ${A}_{n}$ is $\alpha =n$.

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