Abstract
In the present paper we shall show that the rank of the finite field\(\mathbb{F}_{q^n }\) regarded as an\(\mathbb{F}_q\)-algebra has one of the two values 2n or 2n+1 ifn satisfies 1/2q+1<n<1/2(m(q)−2). Herem(q) denotes the maximum number of\(\mathbb{F}_q\)-rational points of an algebraic curve of genus 2 over\(\mathbb{F}_q\). Using results of Davenport-Hasse, Honda and Rück we shall give lower bounds form(q) which are close to the Hasse-Weil bound\(q + 1 + 4\sqrt q\). For specialq we shall further show thatm(q) is equal to the Hasse-Weil bound.
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Shokrollahi, M.A. On the rank of certain finite fields. Comput Complexity 1, 157–181 (1991). https://doi.org/10.1007/BF01272519
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DOI: https://doi.org/10.1007/BF01272519