## Osaka Journal of Mathematics

### A statistical relation of roots of a polynomial in different local fields III

Yoshiyuki Kitaoka

#### Abstract

Let $f(x)$ be a monic polynomial in $\mathbb{Z}[x]$. We have observed a statistical relation of roots of $f(x) \bmod p$ for different primes $p$, where $f(x)$ decomposes completely modulo $p$. We could guess what happens if $f(x)$ is irreducible and has at most one decomposition $f(x) = g(h(x))$ such that $g,h$ are monic polynomials over $\mathbb{Z}$ with $h(0) = 0$, $1 < \deg h < \deg f$. In this paper, we study cases that $f$ has two different such decompositions. Besides, we construct a series of polynomials f which have two non-trivial different decompositions $f(x) = g(h(x))$.

#### Article information

Source
Osaka J. Math., Volume 49, Number 2 (2012), 393-420.

Dates
First available in Project Euclid: 20 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1340197932

Mathematical Reviews number (MathSciNet)
MR2945755

Zentralblatt MATH identifier
1254.11091

#### Citation

Kitaoka, Yoshiyuki. A statistical relation of roots of a polynomial in different local fields III. Osaka J. Math. 49 (2012), no. 2, 393--420. https://projecteuclid.org/euclid.ojm/1340197932

#### References

• Y. Kitaoka: A statistical relation of roots of a polynomial in different local fields, Math. Comp. 78 (2009), 523–536.
• Y. Kitaoka: A statistical relation of roots of a polynomial in different local fields II; in Number Theory, Ser. Number Theory Appl. 6 World Sci. Publ., Hackensack, NJ., 106–126, 2010.