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GroupTheory

 GaloisGroup

 Calling Sequence GaloisGroup( p, x )

Parameters

 p - : polynom : an irreducible polynomial in x over the rationals, or over Q(t1,t2,...,tk), for some indeterminates t1,...,tk. x - : name : an indeterminate

Description

 • The GaloisGroup command returns the Galois group of a polynomial p over the field $ℚ$ of rational numbers, or over the rational function field, $ℚ\left({t}_{1},{t}_{1},\dots ,{t}_{k}\right)$ as a permutation group.  Since the polynomial p is required to be irreducible, the resulting permutation group is transitive.
 • The degree of p may be at most 9 in the univariate case, and at most 8 in the multivariate case.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{GaloisGroup}\left({x}^{5}-x+1,x\right)$
 ${G}{:=}{\mathrm{Gal}}{}\left({{x}}^{{5}}{-}{x}{+}{1}{,}{x}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${120}$ (2)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (3)

Compatibility

 • The GroupTheory[GaloisGroup] command was introduced in Maple 17.