On the construction of the free field

We give a linear algebraic construction of the free field, by constructing the category of "representations": an element of the free field is characterized by the class of objects which can be connected by a chain of morphisms and inverse morphisms; we characterize minimal representations of a given element. We characterize power series and polynomials by their representations among all elements of the free field, and give a primary decomposition of the elements of the free field, extending the classical one for rational functions and that of Fliess for noncommutative rational series. We show that each rational identity in the free field may be "trivialized", that is, is a consequence of the axioms of a field. We give an algorithm for the word problem in the free field, using Grobner bases, different from a previous algorithm of the first author.