Many areas of commutative algebra (e.g., the theory of determinants and the theory of symmetric functions) involve explicit algorithms for computation. Gelfand and Retakh, beginning with their theory of quasideterminants, have obtained noncommutative analogues of many such algorithms. In particular, they have proved a noncommutative version of the Vieta Theorem (which expresses the coefficients of a polynomial in terms of its roots). We refer to such results as examples of "constructive noncommutative algebra."

There is, however, a price to be paid for generalization to the noncommutative case: the Gelfand-Retakh results typically involve rational functions (while polynomials suffice in the commutative case). We discuss a way to avoid working with rational functions: the Gelfand-Retakh version of Vieta's Theorem leads naturally to the definition of a "universal" algebra for solutions of certain polynomial equations. We discuss the structure of this algebra (e.g., grading, basis, Hilbert series) and note that combinatorial objects such as sequences of subsimplices of an n-simplex arise in consideration of these structural questions.