## Hiroshima Mathematical Journal

### A family of entire functions which determines the splitting behavior of polynomials at primes

Hajime Kuroiwa

#### Abstract

In this paper, we prove that there exist entire functions which determines the splitting behavior of polynomials at prime. First, to any monic irreducible polynomial and any prime $p$, we associate a function defined on the set of primes which determines whether the polynomial splits completely at $p$ or not. Then we extend them to entire functions.

#### Article information

Source
Hiroshima Math. J., Volume 41, Number 3 (2011), 409-411.

Dates
First available in Project Euclid: 12 December 2011

https://projecteuclid.org/euclid.hmj/1323700042

Digital Object Identifier
doi:10.32917/hmj/1323700042

Mathematical Reviews number (MathSciNet)
MR2895288

Zentralblatt MATH identifier
1238.11034

Subjects
Primary: 11A41: Primes 11A51: Factorization; primality

#### Citation

Kuroiwa, Hajime. A family of entire functions which determines the splitting behavior of polynomials at primes. Hiroshima Math. J. 41 (2011), no. 3, 409--411. doi:10.32917/hmj/1323700042. https://projecteuclid.org/euclid.hmj/1323700042

#### References

• Y. Ihara; "Koremoaremo...imadatoketeimasen (Neither this nor that is yet solved)" (in Japanese), Suurikagaku (Mathematical science), saiensu-sya, Tokyo, August 1994.
• D. E. Knuth; The Art of Computer Programming volume 2 SEMINUMERICAL ALGORITHMS Arithmetic, Addison-Wesley, 1969.
• J.Neukirch; Algebraische Zahlentheorie, Springer-Verlag, 1992.