## A family of entire functions which determines the splitting behavior of polynomials at primes.(English)Zbl 1238.11034

Author’s abstract: We prove that there exist entire functions which determine the splitting behavior of polynomials at primes. 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.

### MSC:

 11C08 Polynomials in number theory 30D99 Entire and meromorphic functions of one complex variable, and related topics 11R09 Polynomials (irreducibility, etc.)
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### References:

 [1] Y. Ihara; ”Koremoaremo...imadatoketeimasen (Neither this nor that is yet solved)” (in Japanese), Suurikagaku (Mathematical science) , saiensu-sya, Tokyo, August 1994. [2] D. E. Knuth; The Art of Computer Programming volume 2 SEMINUMERICAL ALGORITHMS Arithmetic , Addison-Wesley, 1969. · Zbl 0191.18001 [3] J.Neukirch; Algebraische Zahlentheorie , Springer-Verlag, 1992. · Zbl 0747.11001
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