Translates of completely normal elements and the Morgan-Mullen conjecture

2025/09/27 by Garefalakis, Theodoulos, Kapetanakis, Giorgos
#11T23 #11T30 #12E20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.23245

Abstract

Denote by \mathbb F_q the finite field of order q and by \mathbb F_qⁿ its extension of degree n. Some a∈\mathbb F_qⁿ is called primitive if it generates the multiplicative group \mathbb F_qⁿ^* and it is called qⁿ/q-normal if its \mathbb F_q-conjugates form an \mathbb F_q-basis of \mathbb F_qⁿ if the latter is viewed as an \mathbb F_q-vector space. Furthermore, some a∈\mathbb F_qⁿ is called qⁿ/q-completely normal if it is qⁿ/q^d-normal for all d| n. In this work we prove a new construction of sets of completely normal elements and, we establish, under conditions, the existence of elements that are simultaneously primitive and qⁿ/q-completely normal, covering some yet unresolved cases of a 30-year-old conjecture by Morgan and Mullen.

Citations

• Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields
• Iterated constructions of completely normal polynomials
• Computational Results on the Existence of Primitive Complete Normal Basis Generators
• Primitive Complete Normal Bases for Regular Extensions: Exceptional Cyclotomic Modules
• Further results on the Morgan-Mullen conjecture
• On the existence of primitive completely normal bases of finite fields
• A Class of Incomplete Character Sums

Related

• Further results on the Morgan-Mullen conjecture
• Variations of the Primitive Normal Basis Theorem
• On the existence of primitive completely normal bases of finite fields
• Existence of primitive 1-normal elements in finite fields
• Computational Results on the Existence of Primitive Complete Normal Basis Generators