Sums of primitive roots in a finite field

AbstractLet k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

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Sums of primitive roots of the first and second kind in a finite field

Mathematische Nachrichten ◽

10.1002/mana.19540120303 ◽

1954 ◽

Vol 12 (3-4) ◽

pp. 155-172

Author(s):

Leonard Carlitz

Keyword(s):

Finite Field ◽

Primitive Roots

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Double Character Sums over Subgroups and Intervals

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972714000227 ◽

2014 ◽

Vol 90 (3) ◽

pp. 376-390 ◽

Cited By ~ 5

Author(s):

MEI-CHU CHANG ◽

IGOR E. SHPARLINSKI

Keyword(s):

Finite Field ◽

Upper Bound ◽

Multiplicative Group ◽

Character Sums ◽

Multiplicative Character ◽

Nontrivial Estimate ◽

Image Position ◽

Primitive Roots ◽

Double Character

AbstractWe estimate double sums $$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$ with a multiplicative character ${\it\chi}$ modulo $p$ where ${\mathcal{I}}=\{1,\dots ,H\}$ and ${\mathcal{G}}$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$ can be derived from the Burgess bound if $H\geq p^{1/4+{\it\varepsilon}}$ and from some standard elementary arguments if $T\geq p^{1/2+{\it\varepsilon}}$, where ${\it\varepsilon}>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$ and give an application to primitive roots modulo $p$ with three nonzero binary digits.

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