On Irreducible Polynomials of the Form $b(x^d)$

Palash Sarkar; Shashank Singh

On Irreducible Polynomials of the Form $b(x^d)$

Number Theory 2016-04-29 v1

Authors: Palash Sarkar , Shashank Singh

Abstract

Let $p$ be a prime and $b(x)$ be an irreducible polynomial of degree $k$ over $\mathbb{F}_p$ . Let $d\geq 1$ be an integer. Consider the following question: Is $b(x^d)$ irreducible? We derive necessary conditions for $b(x^d)$ to be irreducible. Further, when the necessary conditions are satisfied, we obtain the probability for $b(x^d)$ to be irreducible.

Keywords

irreducible polynomials finite fields polynomial of form b(x^d) necessary conditions irreducibility probability prime fields

Cite

@article{arxiv.1604.08303,
  title  = {On Irreducible Polynomials of the Form $b(x^d)$},
  author = {Palash Sarkar and Shashank Singh},
  journal= {arXiv preprint arXiv:1604.08303},
  year   = {2016}
}

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