Hi, I was hoping to post this question to to the sage-support group
directly, but my membership has not yet been approved.
I am having trouble implementing complex conjugation in a relative
extension of a cyclotomic field. I create my field in Sage as
follows:
sage: Q3 = CyclotomicField(3)
sage: z3 = Q3.0
sage: x = polygen(Q3,'x')
sage: Q32.<s2> = Q3[sqrt(Q3(2))]
Then I would like to compute the complex conjugate of an element as
follows, but it doesn't work:
sage: (s2*z3).conjugate()
Traceback (most recent call last):
...
NotImplementedError: complex conjugation is not implemented (or
doesn't make sense).
Okay, but just to check, Sage does know how to conjugate elements of
the base field:
sage: z3.conjugate()
-zeta3 - 1
So one thing to try is to explicitly construct the homomorphism.
First I check the generators:
sage: Q32.gens()
(sqrt2, zeta3)
Then I try to construct what I need:
sage: Q32.hom([s2,z3^2])
Traceback (most recent call last):
...
TypeError: images do not define a valid homomorphism
Somewhere in the documentation, it says that you can list all the
automorphisms (not just those that preserve the base field) by using
embeddings(). And indeed, I find the automorphism I need in that
list:
sage: Q32.embeddings(Q32)[3]
Relative number field endomorphism of Number Field in sqrt2 with
defining polynomial x^2 - 2 over its base field
Defn: sqrt2 |--> sqrt2
zeta3 |--> -zeta3 - 1
So can someone please advise me how to construct this in general? Is
this a bug, or just something that is not implemented yet?
Thanks in advance,
Jon
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