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Question: How do you get all the other roots from the Galois group of polynomials and one of the roots?

Question:How do you get all the other roots from the Galois group of polynomials and one of the roots?

jud 155 Maple
polynomial group-theory galois
March 02 2023
2

Suppose I have a polynomial:

poly := x^6 - 3*x + 3

And I know one of its roots and its galois group:

firstroot := [solve](poly, x)[1]:
G := GaloisGroup(poly, x):

Since G act transitively on the all roots. How do I get all the other roots of this polynomial just by the firstroot and G, but not by solving the equation.
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