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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?

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

1 Answer 1

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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.

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