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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
source | link

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} *)