def test_galois_action_on_embeddings(f, gal_deg):
K = NumberField(f, "a")
G_K = K.galois_group()
G_K_emb, to_emb, from_emb, Kgal, embeddings = galois_action_on_embeddings(
G_K)
# test that to_emb and from_emb are isomorphism
assert G_K.hom(G_K) == from_emb * to_emb
assert G_K_emb.hom(G_K_emb) == to_emb * from_emb
assert to_emb.domain() == G_K
assert to_emb.codomain() == G_K_emb
assert from_emb.domain() == G_K_emb
assert from_emb.codomain() == G_K
assert embeddings[0] == G_K.gen(0).as_hom() * embeddings[G_K_emb.gen(0)
(1) - 1]
assert Kgal.degree() == G_K_emb.cardinality() == gal_deg
assert G_K_emb.degree() == len(embeddings) == K.degree()
#!/usr/bin/env sage
from __future__ import print_function
from sage import *
from sage.rings.number_field.number_field import NumberField
#import sys
#print(sys.argv)
if 1:
K = NumberField(x ^ 4 + 1)
r2 = e8 + e8 ^ 7
assert r2**2 == 2
G = K.galois_group()
assert len(G) == 4