2 added 240 characters in body edited Sep 26 '17 at 9:51 Szabolcs 172k1818 gold badges468468 silver badges10021002 bronze badges If your starting point is an explicit number (i.e. it has no symbolic parameters), then you could use MinimalPolynomial and CoefficientList. This works with a Root object ... MinimalPolynomial[Root[1 + 2 #1 + #1^5 &, 1], x] (* 1 + 2 x + x^5 *) CoefficientList[%, x] (* {1, 2, 0, 0, 0, 1} *)  ... or any other algebraic number. MinimalPolynomial[Sqrt[2] + Sqrt[3], x] (* 1 - 10 x^2 + x^4 *) CoefficientList[%, x] (* {1, 0, -10, 0, 1} *)  Do keep in mind that MinimalPolynomial will aim to generate a polynomial with integer coefficients. Thus be aware of results like this: MinimalPolynomial[Root[1 + 2 #1 + Sqrt[2] #1^2 &, 1], x] (* -1 - 4 x - 4 x^2 + 2 x^4 *)  If your starting point is an explicit number (i.e. it has no symbolic parameters), then you could use MinimalPolynomial and CoefficientList. This works with a Root object ... MinimalPolynomial[Root[1 + 2 #1 + #1^5 &, 1], x] (* 1 + 2 x + x^5 *) CoefficientList[%, x] (* {1, 2, 0, 0, 0, 1} *)  ... or any other algebraic number. MinimalPolynomial[Sqrt[2] + Sqrt[3], x] (* 1 - 10 x^2 + x^4 *) CoefficientList[%, x] (* {1, 0, -10, 0, 1} *)  If your starting point is an explicit number (i.e. it has no symbolic parameters), then you could use MinimalPolynomial and CoefficientList. This works with a Root object ... MinimalPolynomial[Root[1 + 2 #1 + #1^5 &, 1], x] (* 1 + 2 x + x^5 *) CoefficientList[%, x] (* {1, 2, 0, 0, 0, 1} *)  ... or any other algebraic number. MinimalPolynomial[Sqrt[2] + Sqrt[3], x] (* 1 - 10 x^2 + x^4 *) CoefficientList[%, x] (* {1, 0, -10, 0, 1} *)  Do keep in mind that MinimalPolynomial will aim to generate a polynomial with integer coefficients. Thus be aware of results like this: MinimalPolynomial[Root[1 + 2 #1 + Sqrt[2] #1^2 &, 1], x] (* -1 - 4 x - 4 x^2 + 2 x^4 *)  1 answered Sep 24 '17 at 14:04 Szabolcs 172k1818 gold badges468468 silver badges10021002 bronze badges If your starting point is an explicit number (i.e. it has no symbolic parameters), then you could use MinimalPolynomial and CoefficientList. This works with a Root object ... MinimalPolynomial[Root[1 + 2 #1 + #1^5 &, 1], x] (* 1 + 2 x + x^5 *) CoefficientList[%, x] (* {1, 2, 0, 0, 0, 1} *)  ... or any other algebraic number. MinimalPolynomial[Sqrt[2] + Sqrt[3], x] (* 1 - 10 x^2 + x^4 *) CoefficientList[%, x] (* {1, 0, -10, 0, 1} *)