Hi John!
On 25 Nov., 10:30, John Cremona <john.crem...@gmail.com> wrote:
> Surely a number field + embedding is a richer structure than an
> abstract number field, so the coercion should go from the former to
> the latter as a forgetful functor.
To the contrary, it seems to me that a coercion should go from the
poorer structure to the richer structure, if there is a canonical way
to do so.
Example: A field is richer than a ring - and we have
sage: GF(5).has_coerce_map_from(Integers(5))
True
sage: Integers(5).has_coerce_map_from(GF(5))
False
For number fields, we *do* have a canonical choice, namely (provided
that the defining polynomials match) sending the generator to the
generator.
Of course, mapping generator to generator would work in both
directions. However, if there is a way to preserve structure (here:
embedding) then one should by all means do so.
> Are you going to similarly come up with the question of coercion
> between two number fields which are the same abstract field and with
> mathematically the same embedding, but with one of higher precision
> than the other?
Let K be a number field and R be some ring.
I think there should be coercion from R to K if
* R coerces into the base ring of K (which might be another number
field)
or
* R is a number field and
+ R.base_ring() coerces into K.base_ring() (again covering stacks
of number fields)
+ R.relative_polynomial() == K.relative_polynomial() and
+ either R has no given embedding or
- R.coerce_embedding()(R.gen())==K.coerce_embedding()(K.gen())
and
- R.coerce_embedding()(R.gen()).parent() coerces into
K.coerce_embedding()(K.gen()).parent()
The last line means that it goes from higher to lower precision, since
we have
sage: RealField(4).has_coerce_map_from(RealField(50))
True
> PS What's wrong with sage-nt?!
I thought this was a question on coercion (of number fields), not on
number fields as such.
But I guess the simple reason is that I don't usually read sage-nt.
Cheers,
Simon
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