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.

We show that the p-operator in the Witt algebra (the restricted Lie algebra of derivations of the quotient of the polynomial algebra over a field of characteristic p by the ideal generated by the p-th power of the indeterminant) is given by multiplication by a scalar.