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Character sums, primitive elements, and powers in finite fields. (English) Zbl 1008.11069
Let $$F_{q}$$ be a finite field of characteristic $$p$$ with $$q=p^{n}$$ elements, and $$F_{q^{m}}=F_{q}( \alpha)$$ its extension of degree $$m \geq 1$$. H. Davenport [On primitive roots in finite fields, Q. J. Math., Oxf. Ser. 8, 308-312 (1937; Zbl 0018.10901)] proved that the set $$F_{q}+ \alpha$$ contains at least one primitive element of $$F_{q^{m}}$$ if $$q$$ is sufficiently large with respect to $$m$$.
The author extends this result to certain subsets of $$F_{q}+ \alpha$$ of cardinality at least of the order of magnitude $$O(q^{1/2+ \varepsilon})$$. The proof is based on a new bound for incomplete character sums $S_{K}( \alpha)= \sum_{k=0}^{K-1} \chi( \alpha+ \zeta_{k}), \qquad 1 \leq K < q^{m},$ where $$\chi$$ is a nontrivial multiplicative character of $$F_{q^{m}}$$ and $$\zeta_{0}, \zeta_{1}, \ldots , \zeta_{K-1}$$ are elements of $$F_{q^{m}}$$ defined as follows: if $$k=k_{1}+k_{2} p+ \cdots +k_{mn} p^{mn-1}$$ is the $$p$$-adic expansion of $$0 \leq k < q^{m}$$ with $$0 \leq k_{i} < p$$, $$\{ \omega_{1}, \ldots , \omega_{n} \}$$ is a basis of $$F_{q}$$ over $$F_{p}$$, and $$\omega_{1}, \ldots , \omega_{mn}$$ its completion to a basis of $$F_{q^{m}}$$ over $$F_{p}$$ then $\zeta_{k}=k_{1} \omega_{1}+ \cdots k_{mn} \omega_{mn}.$ The author shows that $|S_{K}( \alpha)|< 2.2 (Km)^{1/2} q^{1/4},$ for $$1 \leq K < q$$, and deduces that the set $$\{ \alpha, \alpha+ \zeta_{1}, \ldots, \alpha+ \zeta_{K} \}$$, with $$\zeta_{i} \in F_{q}$$, contains at least $\frac{ \varphi(q^{m}-1)}{q^{m}-1}(K+O(K^{1/2}q^{1/4+ \varepsilon}))$ primitive elements of $$F_{q^{m}}$$ for any $$\varepsilon > 0$$. Moreover, the author obtains a new upper bound for the longest sequence $\alpha, \alpha+ \zeta_{1}, \ldots, \alpha+ \zeta_{L-1}$ of consecutive powers in $$F_{q^{m}}$$.

##### MSC:
 11T24 Other character sums and Gauss sums 11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
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