The paper consists of two parts, both related to problems of Lubelski, but unrelated otherwise. Theorem 1 enumerates for a = 1,2 the finitely many positive integers D such that every odd positive integer L that divides x² +Dy² for (x,y) = 1 has the property that either L or is properly represented by x²+Dy². Theorem 2 asserts the following property of finite extensions k of ℚ : if a polynomial f ∈ k[x] for almost all prime ideals of k has modulo at least v linear factors, counting multiplicities, then either f is divisible by a product of v+1 factors from k[x]∖ k, or f is a product of v linear factors from k[x].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba59-2-2,
author = {A. Schinzel},
title = {Solution to a Problem of Lubelski and an Improvement of a Theorem of His},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {59},
year = {2011},
pages = {115-119},
zbl = {1255.11019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba59-2-2}
}
A. Schinzel. Solution to a Problem of Lubelski and an Improvement of a Theorem of His. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011) pp. 115-119. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba59-2-2/