# QQ.extension() with embedding: incorrect modulus

Minimal Example:

print version() # get the version

ZX.<X>=ZZ[] # polynomials in X
f=X^3-X-1 # minimal polynomial of minimal Pisot
roots=f.complex_roots() # its roots
root_beta=roots[0] # the real root is first
print 'A:', root_beta # verify we got the real root

Qb.<ext_beta> = QQ.extension(f, embedding=root_beta) # make an extension
float_beta = CC(ext_beta) # convert to float
print 'B:', float_beta.abs()
print 'C:', ext_beta.abs() # why B != C ???

print 'D:', roots[1].abs() # it turns out that ext_beta.abs() gives the modulus of another root of f !!!


Output:

Sage Version 6.1.1, Release Date: 2014-02-04
A: 1.32471795724475
B: 1.32471795724475
C: 0.868836961832709
D: 0.868836961832709


The problem is that ext_beta.abs() != CC(ext_beta).abs() for ext_beta the generator of an algebraic extension, which just doesn't sound right (and arises some problems of course).

I want to work in precise arithmetics, therefore I need to apply abs() without conversion to floats. Is there any way how to achieve this?

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This is definitely a defect in Sage. If z is an element in a number field K, then z.abs(i) returns the absolute value of the the i-th embedding of K into CC , and if i is omitted, the zero-th embedding is used. Thus any specified embedding used to construct K is ignored. I have raises a ticket on this issue; see http://trac.sagemath.org/ticket/16147.

( 2014-04-11 23:15:27 -0500 )edit

Thanks for reporting, Tom !

( 2014-04-12 00:39:04 -0500 )edit

Thanks @Francis for pointing it out, I wasn't aware that this is the issue! And you're welcome, Thierry :)

( 2014-04-17 02:26:42 -0500 )edit

I provided a fix on the ticket http://trac.sagemath.org/ticket/16147

( 2014-06-29 07:49:35 -0500 )edit

Thanks to Francis for the review, it is in the 6.3.beta5 release now. So it will be there in sage 6.3!

( 2014-07-01 02:48:00 -0500 )edit