  It is well known that over an infinite field the ring of symmetric functions
in a finite number of variables is isomorphic to the one of polynomial
functions on matrices that are invariants by the action of conjugation by
general linear group. We generalize this result showing that the abelianization
of the algebra of the symmetric tensors of fixed order over a free associative
algebra is isomorphic to the algebra of the polynomials invariants of several
matrices over an infinite field or the integers. While proving the main result
we find generators and relations of abelianized divided powers of an algebra
over any commutative ring.
