Swan-like results for binomials and trinomials over finite fields of odd characteristic

Swan (Pac. J. Math. 12:1099–1106, 1962) gives conditions under which the trinomial xn + xk + 1 over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}_{2}}$$\end{document} is reducible. Vishne (Finite Fields Appl. 3:370–377, 1997) extends this result to trinomials over extensions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}_{2}}$$\end{document}. In this work we determine the parity of the number of irreducible factors of all binomials and some trinomials over the finite field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}_{q}}$$\end{document}, where q is a power of an odd prime.