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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,

http://magma.maths.usyd.edu.au/calc/

to determine the Galois group of $x^5+5x-12=0$ (for example)?

Tito Piezas III
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3 Answers3

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> P< x >:=PolynomialAlgebra(Rationals());

> f:=x^5+5*x-12;

> G:=GaloisGroup(f);

> 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.

Community
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Glasby
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  • Please use $\LaTeX$ to format your post. – J.R. Jan 25 '14 at 13:29
  • Someone also recommended this Magma site. The reason is I found a family of solvable equations of deg $p=5,7,13$. I already knew how to solve them in radicals, but I also wanted to explicitly determine their Galois group. – Tito Piezas III Jan 25 '14 at 19:35
  • @Glasby Is there a way of making the base field finite, i.e. calculating the Galois Group of a polynomial over $\mathbb{F}_7$? – Mathmo May 14 '14 at 10:18
  • Your code gives «User error: Identifier 'x' has not been declared or assigned» on Magma site (link to the site is given in OP). Should be P<x>:=…. – Artem Pelenitsyn Feb 27 '16 at 22:40
  • @Mathmo No GaloisGroup works for number field and orders, but finite fields are easy. The Galois group over a finite field of an irreducible polynomial of degree $n$ is a cyclic group of order $n$. – Glasby Aug 03 '21 at 10:39
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For x^10 - 1280/512*x^8 + 1120/512*x^6 - 400/512*x^4 + 50/512*x^2 - 1/512, the web page can return the generator of Galois group:

R < x > := PolynomialRing (RationalField ());
f := x^10 - 1280/512*x^8 + 1120/512*x^6 - 400/512*x^4 + 50/512*x^2 - 
  1/512;
G := GaloisGroup (f);
G;
IsSolvable (G);
Please correct GrammarMistakes
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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.

user187856
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