Letp>2 be a prime. A functionf: GF(p)→GF(p) is planar if for everya∃GF(p)*, the functionf(x+a−f(x) is a permutation ofGF(p). Our main result is that every planar function is a quadratic polynomial. As a consequence we derive the following characterization of desarguesian planes of prime order. IfP is a protective plane of prime orderp admitting a collineation group of orderp2, thenP is the Galois planePG(2,p). The study of such collineation groups and planar functions was initiated by Dembowski and Ostrom  and our results are generalizations of some results of Johnson .
We have recently learned that results equivalent to ours have simultaneously been obtained by Y. Hiramine and D. Gluck.