Solution to a Problem of Lubelski and an Improvement of a Theorem of His

A. Schinzel. "Solution to a Problem of Lubelski and an Improvement of a Theorem of His." Bulletin of the Polish Academy of Sciences. Mathematics 59.2 (2011): 115-119. <http://eudml.org/doc/281274>.

@article{A2011,
abstract = {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 $2^\{a\}L$ 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].},
author = {A. Schinzel},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {binary quadratic form; polynomial modulo a prime ideal},
language = {eng},
number = {2},
pages = {115-119},
title = {Solution to a Problem of Lubelski and an Improvement of a Theorem of His},
url = {http://eudml.org/doc/281274},
volume = {59},
year = {2011},
}

TY - JOUR
AU - A. Schinzel
TI - Solution to a Problem of Lubelski and an Improvement of a Theorem of His
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2011
VL - 59
IS - 2
SP - 115
EP - 119
AB - 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 $2^{a}L$ 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].
LA - eng
KW - binary quadratic form; polynomial modulo a prime ideal
UR - http://eudml.org/doc/281274
ER -