def galois_action_on_embeddings(G_K):
K = G_K.number_field()
Kgal = G_K.splitting_field()
embeddings = K.embeddings(Kgal)
# first a shortcut in the case where G_K is normal in S_d since then it doesn't
# matter for our application since we only care about the image
# of the galois group in S_d
d = K.absolute_degree()
G_K_roots = TransitiveGroup(d, G_K.transitive_number())
if G_K_roots.is_normal(SymmetricGroup(d)):
id_G_K = G_K.Hom(G_K).identity()
return G_K, id_G_K, id_G_K, Kgal, embeddings
# now for the actual computation
permutations = []
for g in G_K.gens():
phi = g.as_hom()
g_perm = Permutation(
[embeddings.index(phi * emb) + 1 for emb in embeddings]).inverse()
permutations.append(g_perm)
G_K_emb = PermutationGroup(permutations, canonicalize=False)
to_emb = G_K.hom(G_K_emb.gens())
from_emb = G_K_emb.hom(G_K.gens())
return G_K_emb, to_emb, from_emb, Kgal, embeddings
