Let  be a prime and let m be a quadratic non-residue in the finite field  In the present note, we introduce the polynomials  and we provide a short elementary proof of their absolute irreducibility. While an irreducible polynomial in one variable over  has at most one zero, the irreducibles  are in a sense at the other end of the spectrum (interpolating the null function), and constitute an useful grab bag of working examples for the students engaged in the transition (often counterintuitive) from the study of the polynomials in one variable over a field to those in two or more variables. Moreover, the expressions  reduce to particularly concrete forms for special classes of primes in arithmetic progressions corresponding to small values of m, being thus useful for students interested in quadratic residues.