  We describe a solution of the word problem in free fields (coming from
non-commutative polynomials over a commutative field) using elementary linear
algebra, provided that the elements are given by minimal linear
representations. It relies on the normal form of Cohn and Reutenauer and can be
used more generally to (positively) test rational identities. Moreover we
provide a construction of minimal linear representations for the inverse of
non-zero elements.
