  In this dissertation, we investigate the cohomology theory of restricted Lie
algebras. The representation theory of restricted Lie algebras is reviewed
including a description of the restricted universal enveloping algebra. In the
case of an abelian restricted Lie algebra, we construct an augmented complex of
free modules over the enveloping algebra that is exact in dimensions less than
p and hence define the cohomology theory of these algebras in dimension less
than p. In the non-abelian case, we explicitly construct cochain spaces for any
coefficient module in dimensions less than 3, and give explicit formulas for
the coboundary operators in these dimensions. The corresponding notions of the
usual algebraic interpretations of ordinary low dimensional cohomology are
defined and we show that our restricted cohomology spaces encode this
information as well.
