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How can I define a homomorphism from a relative number field K (containing F) to some other field L if I know where to send K.gens()?

Example:

F_pol  = x^2-x-1
F      = NumberField(F_pol, 'lam')
K_pol  = x^2 + 4
K      = F.extension(K_pol, 'e')
L      = QQbar
lam_im = QQbar(F_pol.roots()[1][0])
e_im   = QQbar(K_pol.roots()[1][0])

Wrong result:

K.hom([e_im], QQbar, check=False)

What we want (not working):

K.hom([e_im, lam_im], QQbar, check=False)

How can I define a homomorphism from a relative number field K (containing F) to some other field L if I know where to send K.gens()?

Example:

F_pol  = x^2-x-1
F      = NumberField(F_pol, 'lam')
K_pol  = x^2 + 4
K      = F.extension(K_pol, 'e')
L      = QQbar
lam_im = QQbar(F_pol.roots()[1][0])
L(F_pol.roots()[1][0])
e_im   = QQbar(K_pol.roots()[1][0])
L(K_pol.roots()[1][0])

Wrong result:

K.hom([e_im], QQbar, check=False)

What we want (not working):

K.hom([e_im, lam_im], QQbar, check=False)

K.Hom(L)(e_im, F.hom([lam_im], check=False))

x       = PolynomialRing(QQ,'x').gen()
F_pol   = x^3 - x^2 - 2*x + 1
F.<lam> = NumberField(F_pol, 'lam')
D       = 4*lam^2 + 4*lam - 4
K_pol   = x^2 - D
K       = F.extension(K_pol, 'e')
L       = CC
lam_im  = F_pol.roots(L)[2][0]
e_im    = F.hom([lam_im], check=False)(D).sqrt()

K.Hom(L)(e_im, F.hom([lam_im], check=False), check=False)

TypeError: images do not define a valid homomorphism

How can I define a homomorphism from a relative number field K (containing F) to some other field L if I know where to send K.gens()?

Example:

F_pol  = x^2-x-1
F      = NumberField(F_pol, 'lam')
K_pol  = x^2 + 4
K      = F.extension(K_pol, 'e')
L      = QQbar
lam_im = L(F_pol.roots()[1][0])
e_im   = L(K_pol.roots()[1][0])

Wrong result:

K.hom([e_im], QQbar, check=False)

What we want (not working):

K.hom([e_im, lam_im], QQbar, check=False)

A working solution (edit):

K.Hom(L)(e_im, F.hom([lam_im], check=False))

New question/example: What if L is not exact?

x       = PolynomialRing(QQ,'x').gen()
F_pol   = x^3 - x^2 - 2*x + 1
F.<lam> = NumberField(F_pol, 'lam')
D       = 4*lam^2 + 4*lam - 4
K_pol   = x^2 - D
K       = F.extension(K_pol, 'e')
L       = CC
lam_im  = F_pol.roots(L)[2][0]
e_im    = F.hom([lam_im], check=False)(D).sqrt()

K.Hom(L)(e_im, F.hom([lam_im], check=False), check=False)

This gives the error:

TypeError: images do not define a valid homomorphism

How can I define a homomorphism from a relative number field K (containing F) to some other field L if I know where to send K.gens()?

Example:

F_pol  = x^2-x-1
F      = NumberField(F_pol, 'lam')
K_pol  = x^2 + 4
K      = F.extension(K_pol, 'e')
L      = QQbar
lam_im = L(F_pol.roots()[1][0])
e_im   = L(K_pol.roots()[1][0])

Wrong result:

K.hom([e_im], QQbar, check=False)

What we want (not working):

K.hom([e_im, lam_im], QQbar, check=False)

A working solution (edit):

K.Hom(L)(e_im, F.hom([lam_im], check=False))

New question/example: What if L is not exact?

x       = PolynomialRing(QQ,'x').gen()
F_pol   = x^3 - x^2 - 2*x + 1
F.<lam> = NumberField(F_pol, 'lam')
D       = 4*lam^2 + 4*lam - 4
K_pol   = x^2 - D
K       = F.extension(K_pol, 'e')
L       = CC
lam_im  = F_pol.roots(L)[2][0]
e_im    = F.hom([lam_im], check=False)(D).sqrt()

K.Hom(L)(e_im, F.hom([lam_im], check=False), check=False)

This gives the error:

TypeError: images do not define a valid homomorphism