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embeddings in NumberFieldTower?

asked 2014-12-20 22:37:28 -0500

updated 2015-01-13 14:09:05 -0500

FrédéricC gravatar image

I'm trying to do my calculations in a number field tower (or some equivalent), then get the results in real form (or anything that the graphics functions will take). I'm extending the rationals twice.

I've tried constructing the first with NumberField, then using extension, but apparently embedding isn't implemented in extension.

I've tried using NumberFieldTower, but I can't find an equivalent to embedding.

I've tried using NumberField(poly_1, poly_2) or QQ[poly_1, poly_1], but can't figure out how to assign an embedding or something equivalent.

I've considered trying to construct a field homomorphism from the field to the reals, but I don't know sage well enough to figure out if this is possible.

Is there some equivalent way of getting the values that I want?

Thanks!

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answered 2014-12-21 05:21:33 -0500

FrédéricC gravatar image

You can define an iterated extension using a list:

K.<a,b,c> = NumberField([x^2 + 2, x^3 + 3, x^3 + 2])

Note that the last polynomial is used for the first extension, and so on.

Then you can get the list of embeddings and choose one using for instance

embs = K.embeddings(ComplexField(59)); embs
my_emb = embs[0]
my_emb(a**3+b+c)

But the following does not work so far:

 K.<a,b,c> = NumberField([x^2 + 2, x^3 - 3, x^3 - 2], embedding=[sqrt(2)*CDF(I),3**(1/3),2**(1/3)])
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Thanks! Seems to work.

apeirogon gravatar imageapeirogon ( 2014-12-21 22:03:39 -0500 )edit

Could you please upvate and accept my answer (on the left of it) ?

FrédéricC gravatar imageFrédéricC ( 2014-12-23 03:01:47 -0500 )edit

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Asked: 2014-12-20 22:37:28 -0500

Seen: 33 times

Last updated: Dec 21 '14