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

Volume 59 / 2011

A. Schinzel Bulletin Polish Acad. Sci. Math. 59 (2011), 115-119 MSC: Primary 11E16, 11R09. DOI: 10.4064/ba59-2-2

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 $x^2 +Dy^2$ 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 $k[x]$.

Authors

  • A. SchinzelInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland
    e-mail

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