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Transactions of the American Mathematical Society
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1150018-X
PII: S 0002-9947(1993)1150018-X
Article copyright: © Copyright 1993 American Mathematical Society