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.
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