The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

2020/05/19 by Yağmur Çakıroğlu, Çakıroğlu, Yağmur, Oğuz Yayla +3 · 1 citation
Computer Science · Mathematics · #11G20 #11T06 #94A55 #Cellular Automata and Applications #Coding theory and cryptography #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Information Theory (cs.IT) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2005.09402

openalex publication_date 2020/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present the formula for the number of monic irreducible polynomials of degree n over the finite field \mathbb F_q where the coefficients of x^n-1 and x vanish for n≥3. In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements a∈\mathbb F_qⁿ for which Trace(a)=0 and Trace(a^-1)=0. Besides, we apply the formula to give an upper bound on the number of distinct constructions of a family of sequences with good family complexity and cross-correlation measure.

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