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Construction of a primitive normal base of a finite field. (Russian) Zbl 0694.12014

Let \({\mathbb{F}}_{q^ n}\) be a finite field with \(q^ n\) elements, where \(q=p^ m\) and p is a prime. It is known that \({\mathbb{F}}_{q^ n}\) has a normal basis, i.e., an \({\mathbb{F}}_ q\)-basis of the form \(\omega\), \(\omega^ q,...,\omega^{q^{n-1}}\) with \(\omega \in {\mathbb{F}}_{q^ n}\). An element \(\theta\) of \({\mathbb{F}}_{q^ n}\) is called a primitive element if \(\theta,\theta^ 2,...,\theta^{q^ n-1}\) are pairwise distinct. Let \(\theta^ k=\sum^{n-1}_{i=0}u_ i(k)\omega^{q^ i}\) \((k=1,2,....)\). Then \(\theta^{kq^ v}=\sum^{n-1}_{i=0}u_{i- v}(k)\omega^{q^ i}\) \((v=0,1,...,n-1)\). Here we use \(u_ j(k)=u_{j+n}(k)\) for \(j<0\). The authors prove the following theorem: There exist constants \(c_ 1,c_ 2>0\) such that if \(N>\max (c_ 2n \log q\), exp exp(c\({}_ 1\log^ 2 n))\), there is at least one integer in \(1\leq k\leq N\) satisfying \(\det (u_{i-v}(k))\neq 0\) (0\(\leq i,v\leq n- 1)\) and \((k,q^ n-1)=1\).
Reviewer: Wang Yuan

MSC:

11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
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