  Let K be a field of characteristic p>0, and let q be a power of p. We
determine all polynomials f in K[t]\K[t^p] of degree q(q-1)/2 such that the
Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to
PSL_2(q), subject to a certain ramification hypothesis. As a consequence, we
describe all polynomials f in K[t], of degree not a power of p, such that f is
functionally indecomposable over K but f decomposes over an extension of K.
Moreover, except for one ramification setup (which is treated in the companion
paper arxiv:0707.1837), we describe all indecomposable polynomials f in K[t] of
non-p-power degree which are exceptional, in the sense that x-y is the only
absolutely irreducible factor of f(x)-f(y) which lies in K[x,y]. It is known
that, when K is finite, a polynomial f is exceptional if and only if it induces
a bijection on infinitely many finite extensions of K.
