How to convert a polynomial into monic form of a polynomial

The function ResourceFunction["StauduharGaloisGroup"] can get a Galois Group about a monic irreducible integer polynomial. But I want to know the Galois Group about a non-monic polynomial, such as:$$5 x^5 + 6 x + 6$$So how to convert it into a monic form of a polynomial and keeps its Galois Group unchanged by MMA?

• You can use Sage from within Mathematica to compute it. Dec 29, 2022 at 10:58
• @azerbajdzan Can you teach me how to call sage to solve it in MMA?
– yode
Dec 29, 2022 at 11:08
• By the way, what is wrong with using ResourceFunction["StauduharGaloisGroup"][5 x^5 + 6 x + 6, x]? Dec 29, 2022 at 11:20
• @azerbajdzan ResourceFunction["StauduharGaloisGroup"] tell me its Galois Group of the polynomial is $S(5)$, is wrong, actually its Galois Group is $F(5)$
– yode
Dec 29, 2022 at 12:03
• Sage labels it as C5 : C4 or SmallGroup(20,3). Dec 29, 2022 at 12:29

If we hope to keep the Galois Group unchanged, we should find a Algebraic Integer in the current field. We have two method to do this:

sols5 = SolveValues[5 x^5 + 6 x + 6 == 0, x];
n = First[sols5];


First Method

poly = MinimalPolynomial[AlgebraicNumberDenominator[n]*n, x]


3750 + 750 x + x^5

Second Method

poly = MinimalPolynomial[NumberFieldIntegralBasis[n].RandomInteger[1, 5],x]


5400 + 1080 x - 540 x^2 + x^5

Now, the ResourceFunction["StauduharGaloisGroup"] works well.