I need to test if a family of 7th deg and 13 deg equations are solvable. I’m new to Magma, so my apologies, but what would I type in,
to determine the Galois group of $x^5+5x-12=0$ (for example)?
> P< x >:=PolynomialAlgebra(Rationals());
> print G;
Symmetric group G acting on a set of cardinality 5
Order = 120 = 2^3 * 3 * 5
Although the permutation group on [1..Degree($f$)] is permutationally isomorphic to the Galois group, the bijection with the set of roots of your separable irreducible polynomial $f$ is not determined. For more details see the Magma handbook.
GaloisGroup(FldFin, FldFin) is available. Make an extension of F_7 using your polynomial then call GaloisGroup.
GaloisGroup is also available for polynomials over char p function fields. Coerce your polynomial to be over a function field over F_7 and compute the GaloisGroup of that polynomial.