对几乎所有素数取模的有根多项式

IF 0.5 3区数学 Q3 MATHEMATICS Acta Arithmetica Pub Date : 2022-06-27 DOI:10.4064/aa220407-9-7

C. Elsholtz, Benjamin Klahn, Marc Technau

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引用次数: 2

摘要

如果一个单整数多项式具有除有限个素数以外的所有素数的根模，但没有整数根，则称其为例外多项式。我们对所有不可约的单整数多项式$h$进行分类，其中存在一个不可约的单二次多项式$g$，使得乘积$gh$例外。我们构造了所有因子形式为$X^{p}-b$、$p$撇和$b$平方的例外多项式。

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On polynomials with roots modulo almost all primes

Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic quadratic $g$ such that the product $gh$ is exceptional. We construct exceptional polynomials with all factors of the form $X^{p}-b$, $p$ prime and $b$ square free.

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