On Friday, May 9, 2014 8:06:54 AM UTC-7, John Cremona wrote:
>
> 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.
>
You could leave more of the numerical stability checking in the hands of
sage/pari, in the hope some effort is taken there to ensure the reported
precision.
You'll have an embedding of K into L, so you could look at the embedding of
K.0 that you want and select the embeddings of L that put K.0 embedded L
close to the value you want:
#set up an example
K.<a>=NumberField(x^3-2)
KX.<X>=K[]
Lrel.<b>=NumberField(X^2-(a-2))
Labs=Lrel.absolute_field('babs')
#this selects a complex place that extends the one that sends a to aemb:
aemb=a.complex_embeddings()[0]
min((phi for phi in Labs.embeddings(CC)), key = lambda phi:
abs(phi(Labs(Lrel(a)))-aemb))
#alternatively, if you don't want to/can't depend on coercion:
KtoL=Hom(K,Labs)(a.minpoly().roots(Labs,multiplicities=False)[0])
min((phi for phi in Labs.embeddings(CC)), key = lambda phi:
abs(phi(KtoL(a))-aemb))
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