  We offer an axiomatic definition of a differential algebra of generalized
functions over an algebraically closed non-Archimedean field. This algebra is
of Colombeau type in the sense that it contains a copy of the space of Schwartz
distributions. We study the uniqueness of the objects we define and the
consistency of our axioms. Next, we identify an inconsistency in the
conventional Laplace transform theory. As an application we offer a free of
contradictions alternative in the framework of our algebra of generalized
functions. The article is aimed at mathematicians, physicists and engineers who
are interested in the non-linear theory of generalized functions, but who are
not necessarily familiar with the original Colombeau theory. We assume,
however, some basic familiarity with the Schwartz theory of distributions.
