Period Polynomials for $$ {F_{{{p^{2}}}}} $$ of Fixed Small Degree

Let q = p a with p a prime, and е and f be positive integers satisfying еf + 1 = q. Let F q be the finite field of q elements. The Gauss periods of order е satisfy a period polynimial F(x) of degree е over the rational field Q. In the classical case q = p, Gauss showed that F(x) is irreducible over Q and determined the polynomial for е = 2, 3 and 4.Using the methods and results from the theory of cyclotomy, Dickson, Lehmer, Muskat and Whiteman, among others, determined F(x) for е = 5, 6, 8, 10, 12, 16, 24 in the case q = p. For the general case, Myerson determined F(x) for е = 2, 3 and 4, but beyond this very little is explicitly known. Here I determine F(x) explicitly for the case q = p 2 when е divides 8 or12, by applying work of Berndt and Evans on Gauss sums for $$ {F_{{{p^{2}}}}} $$ with characters of order 6, 8 and 12. The results settle an unanswered question concerning the splitting of the middle factor of F(x) when p ≡ 7(mod 24).