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Yves Lignac
May 9, 2014, 8:30:28 AM5/9/14
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Hello all,
I am looking for a way to determine if an embedding $\sigma$ of a number field $L$ into the complex numbers restricts to a given embedding $\tau$ of a subfield $K$ (asking for equality between $\tau$ and $\sigma \circ i$ where $i$ is the embedding of $K$ into $L$ does not work).
More specifically, I have a cubic field $K$ of signature (1,1) and a quadratic extension $L/K$ with $L$ of signature (0,3), and I want to be able to obtain one of the two complex places of $L$ that does not lie above the real place of $K$. I have tried to do this by taking a polynomial for $L$ over $K$, taking one of its (complex) roots at the complex place of $K$ and using create_embedding_from_approx, but this does not work (on my computer at least) because the latter seems to work only with real embeddings
(for completeness, here is the code I used (given the number fields K,L and the embedding i from K to L :
L_over_K.<u,t> = L.relativize(i)
g = L_over_K.relative_polynomial()
coef = g.coefficients()
expo = g.exponents()
Q = K.places() ; q = Q[1]
coef2 = []
for c in coef :
coef2.append(q(c))
h = 0
y = polygen(CC)
for i in range(0,len(coef)) :
h = h + coef2[i] * y^(expo[i])
root = (complex_roots(h)[0][0]).center()
p = create_embedding_from_approx(L, root)
In any case, if there is a way to test divisibility between archimedean places of number fields without using the above uncomfortable way I would be very much happier to use it
I am looking for a way to determine if an embedding $\sigma$ of a number field $L$ into the complex numbers restricts to a given embedding $\tau$ of a subfield $K$ (asking for equality between $\tau$ and $\sigma \circ i$ where $i$ is the embedding of $K$ into $L$ does not work).
More specifically, I have a cubic field $K$ of signature (1,1) and a quadratic extension $L/K$ with $L$ of signature (0,3), and I want to be able to obtain one of the two complex places of $L$ that does not lie above the real place of $K$. I have tried to do this by taking a polynomial for $L$ over $K$, taking one of its (complex) roots at the complex place of $K$ and using create_embedding_from_approx, but this does not work (on my computer at least) because the latter seems to work only with real embeddings
(for completeness, here is the code I used (given the number fields K,L and the embedding i from K to L :
L_over_K.<u,t> = L.relativize(i)
g = L_over_K.relative_polynomial()
coef = g.coefficients()
expo = g.exponents()
Q = K.places() ; q = Q[1]
coef2 = []
for c in coef :
coef2.append(q(c))
h = 0
y = polygen(CC)
for i in range(0,len(coef)) :
h = h + coef2[i] * y^(expo[i])
root = (complex_roots(h)[0][0]).center()
p = create_embedding_from_approx(L, root)
In any case, if there is a way to test divisibility between archimedean places of number fields without using the above uncomfortable way I would be very much happier to use it
. 
John Cremona
May 9, 2014, 11:08:42 AM5/9/14
to SAGE support
I have forwarded your posting to sage-nt which has readers who may help who don't follow sage-support. You are welcome to join sage-nt (for which I am a moderator so if I see your name I'll let you in!)
Sorry not to have actually helped answer your question!
John Cremona
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