#18865: Can't make ring homomorphism from ring of integers to a residue field
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Reporter: robharron | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.8
Component: number | Keywords: Ring of integers, homset,
fields | residue field
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
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It doesn't seem possible to create a ring homomorphism from an order in a
number field to a residue field of the number field. For instance:
{{{
sage: K.<a> = NumberField(x^2-2)
sage: OK = K.ring_of_integers()
sage: P = K.primes_above(3)[0]
sage: kappa = P.residue_field()
sage: abar = kappa.gen()
sage: im = [g.polynomial().change_ring(ZZ)(abar) for g in OK.gens()]
sage: iota = OK.hom(im)
}}}
raises "TypeError: images do not define a valid homomorphism".
Now, if instead you pass "check=False" to OK.hom, you of course get an
iota, but you are unable to evaluate it:
{{{
sage: iota = OK.hom(im, check=False)
sage: iota(K.gen())
}}}
This raises "TypeError: unsupported operand parent(s) for '*': 'Rational
Field' and 'Residue field in abar of Fractional ideal (3)'". I tried being
clever and doing:
{{{
sage: iota(OK(K.gen()))
}}}
but got the same error. Tracing it back, when sage tries to evaluate iota
at an element a, it calls a._im_gens_(kappa, im) and this is totally wrong
for this homset. Rather it is meant for homomorphisms between number
fields. Basically, it looks like we need a function _im_gens_ for
OrderElement types. It should take the element a written out in the basis
given by OK.gens() and replace the basis elements with the element in im.
--
Ticket URL: <http://trac.sagemath.org/ticket/18865>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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