# A primitive normal pair with prescribed prenorm

Source: [A primitive normal pair with prescribed prenorm](<https://www.sota2.com/research/paper/68fa5c001e209131ce9cd534>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-27T05:33:22.966Z

Authors: K. Chatterjee, S.K. Tiwari

Published: 2024-06-05T18:35:41.000Z

## Abstract

For any positive integers $q$, $n$, $m$ with $q$ being a prime power and $n \\geq 5$, we establish a condition sufficient to ensure the existence of a primitive normal pair $(\\epsilon,f(\\epsilon))$ in $\\mathbb{F}\_{q^{n}}$ over $\\mathbb{F}\_{q}$ such that $\\mathrm{PN}\_{q^n/q}(\\epsilon)=a$, where $a\\in\\mathbb{F}\_{q}$ is prescribed. Here $f={f\_{1}}/{f\_{2}}\\in\\mathbb{F}\_{q^n}(x)$ is a rational function subject to some minor restrictions such that deg($f\_{1}$)+deg($f\_{2}$)$=m$ and $\\mathrm{PN}\_{q^n/q}(\\epsilon)   =\\sum\_{i=0}^{n-1}\\Bigg(\\underset{j\\neq i}{\\underset{0\\leq j\\leq n-1}{\\prod\_{}^{}}}\\epsilon^{q^j}\\Bigg)$. Finally, we conclude that for $m=3$, $n\\geq 6$, and $q=7^k$ where $k\\in\\mathbb{N}$, such a pair will exist certainly for all $(q,n)$ except possibly $10$ choices at most.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2406.03571>)

[PDF](<https://arxiv.org/pdf/2406.03571.pdf>)