# zbMATH — the first resource for mathematics

On polynomials of prescribed height in finite fields. (English. Russian original) Zbl 0665.12018
Math. USSR, Sb. 63, No. 1, 247-255 (1989); translation from Mat. Sb., Nov. Ser. 135(177), No. 2, 253-260 (1988).
Let B be a set of integer points in the n-dimensional cube with side h and integer vertices $$B=\{a_ 0,...,a_{n-1})\in {\mathbb{Z}}^ n$$, $$\ell_ i\leq a_ i\leq \ell_ i+h$$, $$i=0,1,...,n-1\}$$, where h is a natural number, $$\ell_ 0,...,\ell_{n-1}$$ are integers, M(B) is a set of the monic polynomials of degree n with coefficients from B: $$M(B)=\{x^ n+a_{n-1}x^{n-1}+...+a_ 0$$, $$(a_ 0,...,a_{n-1})\in B\}$$. Denote by $$T_ n$$ a set of all n-tuples collections $$t=(t_ 1,...,t_ n)$$ of non-negative integers under the condition $$t_ 1+2t_ 2+...+nt_ n=n$$. The number $$t\in T_ n$$ is called a decomposition type of a polynomial $$f\in {\mathbb{F}}_ p[x]$$ (p is a prime, $${\mathbb{F}}_ p$$ is a finite field) if $f(x)=\prod^{n}_{m=1}\prod^{n}_{k=1}f_{m,k}(x),$ where $$f_{m,k}$$ are irreducible polynomials of degree m over the field $${\mathbb{F}}_ p$$, $$m=1,...,n$$, $$k=1,...,t_ m.$$
It is obtained an asymptotic formula for the number of polynomials (from M(B)) whose reduction modulo p has a given decomposition type $$t\in T_ n$$. For any $$\epsilon >0$$ it is nontrivial for $$h\geq p^{n/n+1+\epsilon}$$. This estimate is applied to a study of the decomposition types of the prime p into prime ideals in fields of algebraic numbers of a certain form.
Furthermore, an asymptotic formula for the number of polynomials $$primitive\quad mod p$$ from M(B) and an estimation of the mean value for the number of $$divisors\quad mod\quad p$$ of polynomials from M(B) are obtained.
Reviewer: S.Kotov

##### MSC:
 11T06 Polynomials over finite fields
Full Text: