1Function: galoisfixedfield 2Section: number_fields 3C-Name: galoisfixedfield 4Prototype: GGD0,L,Dn 5Help: galoisfixedfield(gal,perm,{flag},{v=y}): gal being a Galois group as 6 output by galoisinit and perm a subgroup, an element of gal.group or a vector 7 of such elements, return [P,x] such that P is a polynomial defining the fixed 8 field of gal[1] by the subgroup generated by perm, and x is a root of P in gal 9 expressed as a polmod in gal.pol. If flag is 1 return only P. If flag is 2 10 return [P,x,F] where F is the factorization of gal.pol over the field 11 defined by P, where the variable v stands for a root of P. 12Description: 13 (gen, gen, ?small, ?var):vec galoisfixedfield($1, $2, $3, $4) 14Doc: \var{gal} being be a Galois group as output by \tet{galoisinit} and 15 \var{perm} an element of $\var{gal}.group$, a vector of such elements 16 or a subgroup of \var{gal} as returned by galoissubgroups, 17 computes the fixed field of \var{gal} by the automorphism defined by the 18 permutations \var{perm} of the roots $\var{gal}.roots$. $P$ is guaranteed to 19 be squarefree modulo $\var{gal}.p$. 20 21 If no flags or $\fl=0$, output format is the same as for \tet{nfsubfield}, 22 returning $[P,x]$ such that $P$ is a polynomial defining the fixed field, and 23 $x$ is a root of $P$ expressed as a polmod in $\var{gal}.pol$. 24 25 If $\fl=1$ return only the polynomial $P$. 26 27 If $\fl=2$ return $[P,x,F]$ where $P$ and $x$ are as above and $F$ is the 28 factorization of $\var{gal}.pol$ over the field defined by $P$, where 29 variable $v$ ($y$ by default) stands for a root of $P$. The priority of $v$ 30 must be less than the priority of the variable of $\var{gal}.pol$ (see 31 \secref{se:priority}). 32 In this case, $P$ is also expressed in the variable $v$ for compatibility 33 with $F$. Example: 34 35 \bprog 36 ? G = galoisinit(x^4+1); 37 ? galoisfixedfield(G,G.group[2],2) 38 %2 = [y^2 - 2, Mod(- x^3 + x, x^4 + 1), [x^2 - y*x + 1, x^2 + y*x + 1]] 39 @eprog\noindent 40 computes the factorization $x^4+1=(x^2-\sqrt{2}x+1)(x^2+\sqrt{2}x+1)$ 41