Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The number of irreducible factors of a polynomial. I

Authors: Christopher G. Pinner and Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 339 (1993), 809-834
MSC: Primary 11R09; Secondary 12E05
MathSciNet review: 1150018
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Abstract: Let $ F(x)$ be a polynomial with coefficients in an algebraic number field $ k$. We estimate the number of irreducible cyclotomic factors of $ F$ in $ k[x]$, the number of irreducible noncyclotomic factors of $ F$, the number of $ n$th roots of unity among the roots of $ F$, and the number of primitive $ n$th roots of unity among the roots of $ F$. All of these quantities are counted with multiplicity and estimated by expressions which depend explicitly on $ k$, on the degree of $ F$ and height of $ F$, and (when appropriate) on $ n$. We show by constructing examples that some of our results are essentially sharp.

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Article copyright: © Copyright 1993 American Mathematical Society