Primitive Polynomials of the Form g(x)+λ over Finite Fields: Non-Existence Results and Conjectures

Abstract

In this paper, we investigate the existence of primitive polynomials over 𝔽_qⁿ whose constant term is a primitive element of 𝔽_qⁿ. We prove that such polynomials do not exist if q is odd, qⁿ≡3pmod4, and the degree m of the polynomial is odd. In particular, the polynomials f(x)=g(x)+λ, where g(x)∈𝔽_q[x] satisfies g(0)=0 and λ∈𝔽_qⁿ is primitive, cannot be primitive under the same conditions. Further, for the cubic polynomial x³+x²+x+λ, we establish non-existence results in characteristics $2$ and $3$. These results, in particular, provide counterexamples to previously proposed existence conjectures. We also study the family x^p+x+λ over 𝔽_pⁿ. For an odd prime p, we prove that, provided ∑_i=0ⁿ⁻¹(−1)ⁱλ^pⁱ≠0, the polynomial x^p+x+λ is irreducible over 𝔽_pⁿ if and only if n is even. Motivated by this result and supported by computational evidence, we formulate a conjecture concerning the existence of such primitive polynomials.

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BibTeX RIS

BibTeX

@misc{sharma2026primitive,
  title = {{Primitive Polynomials of the Form $g(x)+λ$ over Finite Fields: Non-Existence Results and Conjectures}},
  author = {Avnish K. Sharma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.07262},
  eprint = {2608.07262},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.07262},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Primitive Polynomials of the Form g(x)+λ over Finite Fields: Non-Existence Results and Conjectures
AU  - Avnish K. Sharma
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2608.07262
UR  - https://arxiv.org/abs/2608.07262
AB  - In this paper, we investigate the existence of primitive polynomials over 𝔽_q^n whose constant term is a primitive element of 𝔽_q^n. We prove that such polynomials do not exist if q is odd, q^n≡3pmod4, and the degree m of the polynomial is odd. In particular, the polynomials f(x)=g(x)+λ, where g(x)∈𝔽_q[x] satisfies g(0)=0 and λ∈𝔽_q^n is primitive, cannot be primitive under the same conditions. Further, for the cubic polynomial x^3+x^2+x+λ, we establish non-existence results in characteristics $2$ and $3$. These results, in particular, provide counterexamples to previously proposed existence conjectures. We also study the family x^p+x+λ over 𝔽_p^n. For an odd prime p, we prove that, provided ∑_i=0^n−1(−1)^iλ^p^i≠0, the polynomial x^p+x+λ is irreducible over 𝔽_p^n if and only if n is even. Motivated by this result and supported by computational evidence, we formulate a conjecture concerning the existence of such primitive polynomials.
ER  -

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