Recurrent methods for constructing irreducible polynomials over \(\mathbb F_{q}\) of odd characteristics. (Q1867478)

The author presents methods for constructing irreducible polynomials over finite fields \({\mathbb F}_q\). Among others he proves : Let \(q\) be odd, \(P(x)\neq x\) be an irreducible polynomial of degree \(n\geq 1\) over \({\mathbb F}_q\), and \(ax^2+bx+c\) and \(dx^2+rx+h\) be relatively prime polynomials from \({\mathbb F}_q[x]\), with \(a\) or \(d\) being non-zero and \(r^2\neq 4dh\). Let \(H(a,d):=a^n\) if \(d=0\) and \(H(a,d):=d^nP\left(\frac{a}{d}\right)\) if \(d\neq 0\). Suppose \[ (ah)^2+(cd)^2+acr^2+b^2dh-bcdr-abhr- 2acdh=\delta^2 \] for some \(\delta\neq 0\) from \({\mathbb F}_q\). Then the polynomial \[ F(x)=(H(a,d))^{-1}(dx^2+rx+h)^n P\left(\frac{ax^2+bx+c}{dx^2+rx+h}\right) \] is irreducible over \({\mathbb F}_q\) if and only if the element \[ (r^2-4dh)^nP\left(\frac{br-2(cd+ah-\delta)} {r^2-4hd}\right)P\left(\frac{br-2(cd+ah+\delta)} {r^2-4hd}\right) \] is a non-square in \({\mathbb F}_q\).