  We describe a standard form for the elements in the universal field of
fractions of free associative algebras (over a commutative field). It is a
special version of the normal form provided by Cohn and Reutenauer and enables
the use of linear algebra techniques for the construction of minimal linear
representations (in standard form) for the sum and the product of two elements
(given in standard form). This completes "minimal" arithmetics in free fields
since "minimal" constructions for the inverse are already known. The
applications are wide: linear algebra (over the free field), rational
identities, computing the left gcd of two non-commutative polynomials, etc.
