Can't make ring homomorphism from ring of integers to a residue field

[rharron]

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.

Component: number fields

Keywords: Ring of integers, homset, residue field

Issue created by migration from https://trac.sagemath.org/ticket/18865

[saraedum]

comment:1

Actually, the following works:

iota = OK.hom(kappa)
iota(K.gen(0))
iota(OK.gen(0))

But you can't define the morphism explicitly.