On an irreducibility theorem of I. Schur
We show that the discriminant of the generalized Laguerre polynomial is a non-zero square for some integer pair , with , if and only if belongs to one of 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 over is the alternating group . For example, we establish that for all but finitely many positive integers , the only for which the Galois group of over is is .
In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x 1k + x 2k + … + x gk, where the x i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit...
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...