rad = Root[-1 - #^2 - #^3 + #^4 + #^6 &, 2];
poly = rad[[1]][x]
PolynomialRemainder[poly, x^2 + a x + b, x]
a /. SolveAlways[% == 0, x];
Factor[poly, Extension -> %[[1]]];
x /. Solve[% == 0, x, Algebraics];
ans = Select[%, N[# == rad] &] // First // Simplify

-1 - x^2 - x^3 + x^4 + x^6

-1 + b  - a b  - a^2 b  - a^4 b + b^2 + 3 a^2 b^2 - b^3 + (a - a^2 - a^3 - a^5 + b + 2 a b + 4 a^3 b - 3 a b^2) x

1/12 (8 3^(2/3) (2/(-9 + Sqrt[849]))^(1/3) - 2^(2/3) (3 (-9 + Sqrt[849]))^(1/3) + Sqrt[ 2 (24 + 96 3^(1/3) (2/(-9 + Sqrt[849]))^(2/3) + 2^(1/3) (3 (-9 + Sqrt[849]))^(2/3))])

rad = Root[-1 - #^2 - #^3 + #^4 + #^6 &, 2];
poly = rad[[1]][x]
PolynomialRemainder[poly, x^2 + a x + b, x]
a /. SolveAlways[% == 0, x];
Factor[poly, Extension -> %[[1]]];
x /. Solve[% == 0, x];
ans = Select[%, N[# == rad] &] // First // Simplify

-1 - x^2 - x^3 + x^4 + x^6

-1 + b- a b- a^2 b- a^4 b + b^2 + 3 a^2 b^2 - b^3 + (a - a^2 - a^3 - a^5 + b + 2 a b + 4 a^3 b - 3 a b^2) x

1/12 (8 3^(2/3) (2/(-9 + Sqrt[849]))^(1/3) - 2^(2/3) (3 (-9 + Sqrt[849]))^(1/3) + Sqrt[ 2 (24 + 96 3^(1/3) (2/(-9 + Sqrt[849]))^(2/3) + 2^(1/3) (3 (-9 + Sqrt[849]))^(2/3))])

PolynomialRemainder[-1 - x^2 - x^3 + x^4 + x^6, x^2 + a x + b, x]
a /. SolveAlways[% == 0, x] // RootReduce
Factor[-1 - x^2 - x^3 + x^4 + x^6, Extension -> Take[%, 2]] // ToRadicals

rad = Root[-1 - #^2 - #^3 + #^4 + #^6 &, 2];
poly = rad[[1]][x]
PolynomialRemainder[poly, x^2 + a x + b, x]
a /. SolveAlways[% == 0, x];
Factor[poly, Extension -> %[[1]]];
x /. Solve[% == 0, x, Algebraics];
ans = Select[%, N[# == rad] &] // First // Simplify

-1 - x^2 - x^3 + x^4 + x^6

-1 + b - a b - a^2 b - a^4 b + b^2 + 3 a^2 b^2 - b^3 + (a - a^2 - a^3 - a^5 + b + 2 a b + 4 a^3 b - 3 a b^2) x

1/12 (8 3^(2/3) (2/(-9 + Sqrt[849]))^(1/3) - 2^(2/3) (3 (-9 + Sqrt[849]))^(1/3) + Sqrt[ 2 (24 + 96 3^(1/3) (2/(-9 + Sqrt[849]))^(2/3) + 2^(1/3) (3 (-9 + Sqrt[849]))^(2/3))])