It looks like the matrix scheme can't handle situations that involve more than one root of a polynomial. If I've identified the three roots of a three-real-root cubic, say p,q,r with p<q<r, I can write p+2q and distinguish it from q+2p or p+2r etc. I think the matrix method will require determining an extension field where the roots are distinguishable. Rich ------ Quoting Dan Asimov <dasimov@earthlink.net>:
Of course, for degree ? 5, not all algebraic numbers are expressible in terms of iterated +/-/*/÷ and rational powers of integers. This discussion mentions some interesting related issues, in particular that matrices can be used to express algebraic numbers: <https://math.stackexchange.com/questions/657168/understanding-non-solvable-algebraic-numbers>.
?Dan