Some subgroups of a finite field and their applications for obtaining\n explicit factors

2018/06/26 by Manjit Singh, Singh, Manjit
Computer Science · Mathematics · #11T05 #11T55 #12E10 #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1806.11052

openalex publication_date 2018/06/26 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28

Abstract

Let \S_q denote the group of all square elements in the\nmultiplicative group mathbbF_q^* of a finite field mathbbF_q of odd\ncharacteristic containing q elements. Let \O_q be the set of all\nodd order elements of mathbbF_q^*. Then \O_q turns up as a\nsubgroup of \S_q. In this paper, we show that\n\O_q=\⟨4\⟩ if q=2t+1 and, \O_q=\⟨\nt\⟩ if q=4t+1, where q and t are odd primes. This paper also gives\na direct method for obtaining the coefficients of irreducible factors of\nx^2nt-1 in mathbbF_q[x] using the information of generator elements\nof \S_q and \O_q, when q and t are odd primes such\nthat q=2t+1 or q=4t+1.\n

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