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

A. Schinzel¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011) no. 2, pp. 115-119.

Résumé

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 $x^2 +Dy^2$ for $(x,y)=1$ has the property that either $L$ or $2^aL$ is properly represented by $x^2+Dy^2$. Theorem 2 asserts the following property of finite extensions $k$ of $\mathbb{Q}$: if a polynomial $f \in k[x]$ for almost all prime ideals $\mathfrak{p}$ of $k$ has modulo $\mathfrak{p}$ at least $v$ linear factors, counting multiplicities, then either $f$ is divisible by a product of $v+1$ factors from $k[x]\setminus k$, or $f$ is a product of $v$ linear factors from $k[x]$.

Détail

DOI : 10.4064/ba59-2-2

Keywords: paper consists parts related problems lubelski unrelated otherwise theorem enumerates finitely many positive integers every odd positive integer divides has property either properly represented theorem asserts following property finite extensions mathbb polynomial almost prime ideals mathfrak has modulo mathfrak least linear factors counting multiplicities either divisible product factors setminus product linear factors