On Irreducible Polynomials of the Form $b(x^d)$
Palash Sarkar, Shashank Singh

TL;DR
This paper investigates when polynomials of the form $b(x^d)$, derived from irreducible polynomials over finite fields, remain irreducible, providing necessary conditions and probability estimates.
Contribution
It establishes necessary conditions for the irreducibility of $b(x^d)$ and calculates the probability of irreducibility when these conditions are met.
Findings
Necessary conditions for irreducibility of $b(x^d)$ derived
Probability estimates for irreducibility provided
Conditions depend on properties of $b(x)$ and $d$
Abstract
Let be a prime and be an irreducible polynomial of degree over . Let be an integer. Consider the following question: Is irreducible? We derive necessary conditions for to be irreducible. Further, when the necessary conditions are satisfied, we obtain the probability for to be irreducible.
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Taxonomy
TopicsMathematical functions and polynomials · Mathematics and Applications · Differential Equations and Numerical Methods
