  In this paper we give an effective criterion as to when a positive integer q
is the order of an automorphism of a smooth hypersurface of dimension n and
degree d, for every d>2, n>1, (n,d)\neq (2,4), and \gcd(q,d)=\gcd(q,d-1)=1.
This allows us to give a complete criterion in the case where q=p is a prime
number. In particular, we show the following result: If X is a smooth
hypersurface of dimension n and degree d admitting an automorphism of prime
order p then p<(d-1)^{n+1}; and if p>(d-1)^n then X is isomorphic to the Klein
hypersurface, n=2 or n+2 is prime, and p=\Phi_{n+2}(1-d) where \Phi_{n+2} is
the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to
intermediate jacobians of Klein hypersurfaces.
