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On the rank of certain finite fields. (English) Zbl 0786.14011

Let \(L\) and \(K\) be two fields with \(L\) a simple extension of \(K\). By \(R(L/K)\) the author means the rank of the tensor associated to multiplication in \(L\) viewed as an \(K\)-algebra. In this paper the author shows that \(R(\mathbb{F}_{q^ n}/\mathbb{F}_ q)\) has one of two possible values when \({1\over 2}q+1<n<{1\over 2}(m(q)-2)\) where \(m(q)\) is the maximum number of \(\mathbb{F}_ q\)-rational points on an algebraic curve of genus 2 over \(\mathbb{F}_ q\). Lower bounds for \(m(q)\) are given which are close to the bound given by Hasse and Weil and for special \(q\) it is shown that \(m(q)\) is the Hasse-Weil bound.

MSC:

14G15 Finite ground fields in algebraic geometry
14H05 Algebraic functions and function fields in algebraic geometry
11T55 Arithmetic theory of polynomial rings over finite fields
14G05 Rational points
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