## The number of irreducible factors of a polynomial. I

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- by Christopher G. Pinner and Jeffrey D. Vaaler PDF
- Trans. Amer. Math. Soc.
**339**(1993), 809-834 Request permission

## 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.## References

- Tom M. Apostol,
*Resultants of cyclotomic polynomials*, Proc. Amer. Math. Soc.**24**(1970), 457–462. MR**251010**, DOI 10.1090/S0002-9939-1970-0251010-X - Paul T. Bateman,
*The distribution of values of the Euler function*, Acta Arith.**21**(1972), 329–345. MR**302586**, DOI 10.4064/aa-21-1-329-345 - Enrico Bombieri,
*Lectures on the Thue principle*, Analytic number theory and Diophantine problems (Stillwater, OK, 1984) Progr. Math., vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 15–52. MR**1018368** - Enrico Bombieri and Jeffrey D. Vaaler,
*Polynomials with low height and prescribed vanishing*, Analytic number theory and Diophantine problems (Stillwater, OK, 1984) Progr. Math., vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 53–73. MR**1018369** - D. C. Cantor and E. G. Straus,
*On a conjecture of D. H. Lehmer*, Acta Arith.**42**(1982/83), no. 1, 97–100. MR**679001**, DOI 10.4064/aa-42-1-97-100 - K. Chandrasekharan,
*Introduction to analytic number theory*, Die Grundlehren der mathematischen Wissenschaften, Band 148, Springer-Verlag New York, Inc., New York, 1968. MR**0249348** - E. Dobrowolski,
*On a question of Lehmer and the number of irreducible factors of a polynomial*, Acta Arith.**34**(1979), no. 4, 391–401. MR**543210**, DOI 10.4064/aa-34-4-391-401 - Emma T. Lehmer,
*A numerical function applied to cyclotomy*, Bull. Amer. Math. Soc.**36**(1930), no. 4, 291–298. MR**1561938**, DOI 10.1090/S0002-9904-1930-04939-3 - Roland Louboutin,
*Sur la mesure de Mahler d’un nombre algébrique*, C. R. Acad. Sci. Paris Sér. I Math.**296**(1983), no. 16, 707–708 (French, with English summary). MR**706663** - Ulrich Rausch,
*On a theorem of Dobrowolski about the product of conjugate numbers*, Colloq. Math.**50**(1985), no. 1, 137–142. MR**818097**, DOI 10.4064/cm-50-1-137-142
A. Schinzel, - A. Schinzel,
*On the number of irreducible factors of a polynomial. II*, Ann. Polon. Math.**42**(1983), 309–320. MR**728089**, DOI 10.4064/ap-42-1-309-320 - C. J. Smyth,
*On the product of the conjugates outside the unit circle of an algebraic integer*, Bull. London Math. Soc.**3**(1971), 169–175. MR**289451**, DOI 10.1112/blms/3.2.169 - J. T. Tate,
*Global class field theory*, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 162–203. MR**0220697** - A. Zygmund,
*Trigonometric series: Vols. I, II*, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR**0236587**

*On the number of irreducible factors of a polynomial*, Colloq. Math. Soc. János Bolyai, Debrecen (Hungary), 1974.

## Additional Information

- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**339**(1993), 809-834 - MSC: Primary 11R09; Secondary 12E05
- DOI: https://doi.org/10.1090/S0002-9947-1993-1150018-X
- MathSciNet review: 1150018