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

A. Schinzel

Bulletin of the Polish Academy of Sciences. Mathematics (2011)

  • Volume: 59, Issue: 2, page 115-119
  • ISSN: 0239-7269

Abstract

top
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].

How to cite

top

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 -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.