Recursive constructions of irreducible polynomials over finite fields (Q439081)

Let \(F_q\) be a finite field of characteristic \(p\) and let \(P(x)\in F_q[x]\) be an irreducible polynomial of degree \(n\geq 2\). The authors give conditions for the irreducibility of NEWLINE\[NEWLINE F(x) = (x^p-bx+h)^nP\left(\frac{x^p-bx+c}{x^p-bx+h}\right). NEWLINE\]NEWLINE As main tool they use a result in [\textit{S. D. Cohen}, Proc. Camb. Philos. Soc. 66, 335--344 (1969; Zbl 0177.06601)] on the irreducibility of polynomials of the form \(g^n(x)P(f(x)/g(x))\). Recursively, the authors obtain irreducible polynomials of higher degree.