PERFECT POLYNOMIALS REVISITED

Earlier it was shown that every splitting polynomial A = = (x(p) - x)Np(n)-1 with N\(p - 1), n greater-than-or-equal-to 0 is perfect over GF(p); i.e., the sum sigma-(A) of the distinct monic divisors of A over GF(p) equals A. Conversely, it was proved in detail that whenever a splitting polynomial A = [GRAPHICS] is perfect over GF(p) then the N(i)\(p - 1) and n(0) = ... = n(p - 1); and it was claimed (as already proved by CANADAY for p = 2) that N(0) = ... = N(p - 1). This note verifies the claim in detail, via an argument on the level divisors of A. In the process, an equivalence relation is exhibited on the set of splitting perfect polynomials over GF(p) and an intriguing multinomial identity modulo p is discovered.