Function: nfgaloisapply
Section: number_fields
C-Name: galoisapply
Prototype: GGG
Help: nfgaloisapply(nf,aut,x): apply the Galois automorphism aut to the object
x (element or ideal) in the number field nf.
Doc: let $\var{nf}$ be a
number field as output by \kbd{nfinit}, and let \var{aut} be a \idx{Galois}
automorphism of $\var{nf}$ expressed by its image on the field generator
(such automorphisms can be found using \kbd{nfgaloisconj}). The function
computes the action of the automorphism \var{aut} on the object $x$ in the
number field; $x$ can be a number field element, or an ideal (possibly
extended). Because of possible confusion with elements and ideals, other
vector or matrix arguments are forbidden.
\bprog
? nf = nfinit(x^2+1);
? L = nfgaloisconj(nf)
%2 = [-x, x]~
? aut = L[1]; /* the non-trivial automorphism */
? nfgaloisapply(nf, aut, x)
%4 = Mod(-x, x^2 + 1)
? P = idealprimedec(nf,5); /* prime ideals above 5 */
? nfgaloisapply(nf, aut, P[2]) == P[1]
%6 = 0 \\ !!!!
? idealval(nf, nfgaloisapply(nf, aut, P[2]), P[1])
%7 = 1
@eprog\noindent The surprising failure of the equality test (\kbd{\%7}) is
due to the fact that although the corresponding prime ideals are equal, their
representations are not. (A prime ideal is specified by a uniformizer, and
there is no guarantee that applying automorphisms yields the same elements
as a direct \kbd{idealprimedec} call.)
The automorphism can also be given as a column vector, representing the
image of \kbd{Mod(x, nf.pol)} as an algebraic number. This last
representation is more efficient and should be preferred if a given
automorphism must be used in many such calls.
\bprog
? nf = nfinit(x^3 - 37*x^2 + 74*x - 37);
? aut = nfgaloisconj(nf)[2]; \\ @com an automorphism in basistoalg form
%2 = -31/11*x^2 + 1109/11*x - 925/11
? AUT = nfalgtobasis(nf, aut); \\ @com same in algtobasis form
%3 = [16, -6, 5]~
? v = [1, 2, 3]~; nfgaloisapply(nf, aut, v) == nfgaloisapply(nf, AUT, v)
%4 = 1 \\ @com same result...
? for (i=1,10^5, nfgaloisapply(nf, aut, v))
time = 463 ms.
? for (i=1,10^5, nfgaloisapply(nf, AUT, v))
time = 343 ms. \\ @com but the latter is faster
@eprog