0 A Note on a Specific Pencil of Conics in the Galois Fields of Order

Our remark concerns the structure of the pencil of conics containing three degenerate conics of which two represent conjugate imaginary line pairs, i.e. the pencil defined by Eq. (24) in [1]. In what follows we will demonstrate that Campbell’s claim that also the third singular conic ‘must be a conjugate imaginary line pair’ is false, because this conic is, in fact, a real line pair. The proof relies on the following theorem: the expression u + v +Θuv, Θ 6= 0, (1) where u and v are regarded as variables and Θ is a parameter, is reducible or irreducible in the Galois field of order 2n (GF(2n) iff, respectively, D (