  The concept of descent algebras over a field of characteristic zero is
extended to define descent algebras over a field of prime characteristic. Some
basic algebraic structure of the latter, including its radical and irreducible
modules, is then determined. The decomposition matrix of the descent algebras
of Coxeter group types $A$, $B$, and $D$ are calculated, and used to derive a
description of the decomposition matrix of an arbitrary descent algebra. The
Cartan matrix of a variety of descent algebras over a finite field is then
obtained.
