The galois command computes the Galois group of a polynomial f. The argument f must be an irreducible polynomial in x over Q --the field of rational numbers-- or over $Q\[\mathrm{t1},\mathrm{t2},...,\mathrm{tn}\]$, where the ti's are variables.

The Galois group of a polynomial over $C_t$, where $C$ is the field of complex numbers, can be computed with algcurves[monodromy](f, t, x, group).

For polynomials over Q, the maximum degree is 9. For polynomials over $Q\[\mathrm{t1},\mathrm{t2},...,\mathrm{tn}\]$, the maximum degree in x is 8.

If f is a univariate polynomial, the argument x is optional.

The galois function returns an expression sequence of five elements:

1- A string giving the name of the Galois group using the notation of "The Transitive Groups of Degree up to Eleven" by G. Butler and J. McKay, Communications in Algebra, 11(8) 1983. For instance "8T24" stands for the twenty-fourth group in the list of degree 8 transitive groups.

2- A set of strings, each string giving a description for the group according to "On Transitive Permutation Groups" by J. H. Conway, A. Hulpke and J. McKay, London Mathematical Society Journal of Computation and Mathematics. See group/transnames for more details.

3- A string indicating the parity of the group ("+" for even groups, "-" for odd groups).

4- The order of the group.

5- A set of generators in disjoint cycle notation. The generators are represented by strings.

Note that specialization of the parameters t1, t2, ... ,tn results in a group which is a subgroup of the original group. (See, for example, van der Wärden, Algebra I, 8.10.)

The set of generators can be used to build a permutation group with the permgroup command of the group package.

The function group[transgroup] returns information about transitive permutation groups.

The galois procedure is able to print out details of how the Galois group is computed.  To see some of this information you must first type infolevel[galois]:=1; before calling galois. To see all of the details type infolevel[galois]:=2; before invoking galois.

$\mathrm{galois}\left(x^4+x+1\right)$

$''4T5'',\left\{''S(4)''\right\},''-'',24,\left\{''(1\; 4)''\,''(2\; 4)''\,''(3\; 4)''\right\}$

$\mathrm{galois}\left(t^5-5t+12\right)$

$''5T2'',\left\{''5:2''\,''D(5)''\right\},''+'',10,\left\{''(1\; 2\; 3\; 4\; 5)''\,''(1\; 4)(2\; 3)''\right\}$

$G≔\mathrm{galois}\left(x^5+2\right)$

$G≔''5T3'',\left\{''5:4''\,''F(5)''\right\},''-'',20,\left\{''(1\; 2\; 3\; 4\; 5)''\,''(1\; 2\; 4\; 3)''\right\}$

$\mathrm{with}\left(\mathrm{group}\right)\:$

$\mathrm{DerivedS}\left(\mathrm{permgroup}\left(5\,G\left[5\right]\right)\right)$

$\left[\mathrm{permgroup}\left(5\,\left\{\left[\left[1\,2\,4\,3\right]\right]\,\left[\left[1\,2\,3\,4\,5\right]\right]\right\}\right)\,\mathrm{permgroup}\left(5\,\left\{\left[\right]\,\left[\left[1\,2\,3\,4\,5\right]\right]\right\}\right)\,\mathrm{permgroup}\left(5\,\left\{\left[\right]\right\}\right)\right]$

$\mathrm{galois}\left(x^7+4x^5-3x^2+5\right)$

$''7T7'',\left\{''S(7)''\right\},''-'',5040,\left\{''(1\; 7)''\,''(2\; 7)''\,''(3\; 7)''\,''(4\; 7)''\,''(5\; 7)''\,''(6\; 7)''\right\}$

$\mathrm{galois}\left(x^8-8x^6-8x^5+8\right)$

$''8T45'',\left\{''[1/2S(4)^2]2''\right\},''+'',576,\left\{''(1\; 2\; 3)''\,''(1\; 8)(4\; 5)''\,''(2\; 8)(1\; 3)''\,''(4\; 8)(1\; 5)(2\; 6)(3\; 7)''\right\}$

$\mathrm{galois}\left({\left(x^4-y\right)}^2+1\,x\right)$

$''8T17'',\left\{''[4^2]2''\right\},''-'',32,\left\{''(1\; 2\; 3\; 8)''\,''(4\; 8)(1\; 5)(2\; 6)(3\; 7)''\right\}$

$\mathrm{galois}\left(t^3+\mathrm{x1}{t}^2+\mathrm{x2}t+\mathrm{x3}\,t\right)$

$''3T2'',\left\{''S(3)''\right\},''-'',6,\left\{''(1\; 3)''\,''(2\; 3)''\right\}$

$\mathrm{galois}\left(x^9-9x^7+27x^5-30x^3+9x-1\right)$

$''9T1'',\left\{''C(9)''\right\},''+'',9,\left\{''(1\; 2\; 3\; 4\; 5\; 6\; 7\; 8\; 9)''\right\}$