Factorization of trinomials over Galois fields of characteristic 2
Let \(r\) denote the number of monic irreducible factors of a monic \(f \in K[x]\), where \(K\) is a field and \(f\) has degree \(n\). If \(f=x^n+ax^k+b\) (with \(k<n\)), then \(f\) is called a trinomial. Using a characterization for the parity of \(n-r\) in terms of the discriminant of \(f\) (due to \textit{L. Stickelberger} [Verh. Int. Math. Congr. 1, 182--193 (1897; JFM 29.0172.03)]) in combination with a formula for the discriminant of a trinomial (due to \textit{R. G. Swan} [Pac. J. Math. 12, 1099--1106 (1962; Zbl 0113.01701)]), the author provides characterizations for the parity of \(n-r\) or \(r\) in the case where \(K\) is a Galois field of characteristic \(2\). In particular, reducibility criteria are obtained, e.g., if \([K: \text{GF}(2)]\) is even and \(n\) is even, then \(f\) is reducible, except possibly when \(f=x^{2d}+ax^d+b\) and \(x^2+ax+b\) is irreducible over \(K\).
scientific article; zbMATH DE number 1199968