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I am trying to understand the Galois group of splitting fields (over $\mathbb Q$) of some polynomials. Especially those for which the Galois group is not the full symmetric group. For instance the splitting field of $x^6-x^5+2x^4+x^3+2x^2+3x+1$ (Galois group isomorphic to $S_4$) or of $x^6 - x^5 + x^4 - x^3 - 4x^2 + 5$ (Galois group isomorphic to a transitive group of order $36$). In Pari/GP there is the command polgalois. But this just gives me what the Galois group is isomorphic to. I am also interested in the ordering of the root used for the polynomial input. Magma apparently gives me a bit more, at least this is my impression after reading a bit of its documentation here.

Magma gives an example from its documentation which I quote here

> Z:= Integers();
> P<x>:= PolynomialRing(Z);
> G, R, S := GaloisGroup(x^6-108);
> G;
Permutation group G acting on a set of cardinality 6
Order = 6 = 2 * 3
    (1, 5, 3)(2, 6, 4)
    (1, 2)(3, 6)(4, 5)
> R;
[ -58648*$.1 + 53139 + O(11^5), 58648*$.1 - 19478 +
O(11^5), -43755*$.1 - 72617 + O(11^5), 58648*$.1 -
    53139 + O(11^5), -58648*$.1 + 19478 + O(11^5),
    43755*$.1 + 72617 + O(11^5) ]
> S;
GaloisData over Z_11
> time G, _, S := GaloisGroup(x^32-x^16+2);
Time: 65.760
> #G;
2048

It seems the ordering of the root (for the above its the root of $x^6-108$) is given in R. But I cannot understand R here, it seems to be an encoding for an algebraic number. Is it correct for me to assume that the permutation group is given by the order given by R? If so, how do I read R as roots of $x^6-108$?

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1 Answer 1

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Yes, according to the Magma documentation you linked

Along with the Galois group of a splitting field for f, the roots of f, scaled to be algebraic integers, in a local splitting field and a GaloisData structure are returned.

These roots are given as a sequence (so ordered) and I think it is safe to assume that the ordering of the roots corresponds to the order used in the permutation representation of the Galois group.

If you need to know how to interpret a Magma object, you can ask for the Type(r), Parent(r), or Universe(R) (here R is as defined above, and r is an element of the sequence) for some more detailed information.

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