Factorization of the Cyclotomic Polynomialx2n+ 1 over Finite Fields

The aim of this note is to show that the (well-known) factorization of the 2n+1th cyclotomic polynomialx2n+ 1 over GF(q) withq≡ 1 (mod 4) can be used to prove the (more complicated) factorization of this polynomial over GF(q) withq≡ 3 (mod 4).