  In this paper we investigate the relation between the multiplicities of split
abelian chief factors of finite-dimensional Lie algebras and first degree
cohomology. In particular, we obtain a characterization of modular solvable Lie
algebras in terms of the vanishing of first degree cohomology or in terms of
the multiplicities of split abelian chief factors. The analogues of these
results are well known in the modular representation theory of finite groups.
An important tool in the proof of these results is a refinement of a
non-vanishing theorem of Seligman for the first degree cohomology of
non-solvable finite-dimensional Lie algebras in prime characteristic. As
applications we derive several results in the representation theory of
restricted Lie algebras related to the principal block and the projective cover
of the trivial irreducible module of a finite-dimensional restricted Lie
algebra. In particular, we obtain a characterization of solvable restricted Lie
algebras in terms of the second Loewy layer of the projective cover of the
trivial irreducible module.
