Constructions of irreducible polynomials over finite fields with even characteristic (Q6114139)

The irreducible polynomials over finite fields are an important area of study in Algebra because of their theoretical and practical applications in Cryptography, Coding theory and Computer Algebra Systems. For \(\alpha \in \mathbb{F}_{q}^{n}\) , the \textbf{trace} function \(\operatorname{Tr}_{q^{n} |q}(\alpha)\) of \(\alpha\) over \(\mathbb{F}_{q}\) is defined by \[ \operatorname{Tr}_{q^{n} |q}(\alpha)= \alpha + \alpha^{q} +\cdots +\alpha^{q^{n-1}}. \] Let \(f (x)\) be a polynomial of degree \(n\) in \(\mathbb{F}_{q}\). Then the \textbf{reciprocal} polynomial \(f ^{*}(x)\) of \(f (x)\) is defined by \[ f ^{*}(x)=x^{n}f(\dfrac{1}{x}). \] Main results: \begin{itemize} \item[1.] Let \(q = 2^{s}\) and \(p(x) =\sum_{i=0}^{n} p^{i} x^{i} \) be an irreducible polynomial of degree \(n\) over \(\mathbb{F}_{q}\) . Consider that \(\operatorname{Tr}_{q |2}\left(\dfrac{a p_{n-1}}{b^{2}p_{n}}\right) \neq 0\) and \(\operatorname{Tr}_{q |2}\left(\dfrac{bp_{1}}{ap_{0}}\right)\neq 0\) , where \( a, b \in \mathbb{F}^{*}_{q}\) . Then the polynomial \[ P(x)=(a(x^{2}+x+1))^{n}.p\left(\dfrac{1+b^{2}(x^{2}+x+1)^{2}}{a(x^{2}+x+1)}\right) \] is an irreducible polynomial of degree \(4n\) over \(\mathbb{F}_{q}\). \item[2.] Let \(q = 2^{s}\) and \(p(x) =\sum_{i=0}^{n} p^{i} x^{i} \) be an irreducible polynomial of degree \(n\) over \(\mathbb{F}_{q}\). Consider that \(\operatorname{Tr}_{q |2}\left(\dfrac{a p_{n-1}}{p_{n}}\right) \neq 0\) and \(\operatorname{Tr}_{q |2}\left(\dfrac{bp_{1}}{ap_{0}}\right)\neq 0\), where \( a, b \in \mathbb{F}^{*}_{q}\). Then the polynomial \[ P(x)=(a(x^{4}+x^{3}+x^{2}))^{n}.p\left(\dfrac{(1+b^{2})(x^{4}+x^{2}+1}{a(x^{4}+x^{3}+x^{2})}\right) \] is an irreducible polynomial of degree \(4n\) over \(\mathbb{F}_{q}\). \end{itemize} There are three more results similar to the above results. Examples are also given to demonstrate those five results based on the same field \(\mathbb{F}_{2^{3}}\). The authors could have used different fields for the examples. This article has been authored by five authors and therefore the contribution of each author is considerably small. Also, all theorems are extensions of some existing theorems. Still, there is some room for further extensions of the results over different fields.