A concrete matrix field description of some Galois fields

We provide a concrete description of some factor rings of the polynomial ring k[X], k a field, as fields of matrices, namely of the factor rings k[X]/(Xn − X − 1) for which n is such that the polynomial Xn − X − 1 is irreducible in k[X]. These factor rings include the Galois fields GF(pn) for which Xn − X − 1 is irreducible in Zp[X]. Both MAGMA and MAPLE show that there are many such fields.