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Root[{f, approx}] format was meant for representing roots of non-polynomial functions. Mathematica will never produce algebraic numbers (of degree <= $MaxRootDegree) represented in this way. If you manually enter Root[{f, approx}] with a polynomial f and a correct approximation the result is a valid exact number, but it is not recognized as an algebraic ...


I do not know what is going awry and have sent a question to the relevant person here. I can suggest an alternative route. Form the Root objects using say Solve, figure out which are the ones you want, and use those. p1 = x^5 + 6*x^4 - 42*x^3 - 142*x^2 + 467*x + 422; p2 = Expand[p1 /. {x -> ((x - 1)^2)}]; rts = RootReduce[x /. Solve[p2 == 0, x]]; N[rts] ...

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