#5842: [with patch, needs review] Various number field improvements
---------------------------+------------------------------------------------
Reporter: fwclarke | Owner: was
Type: enhancement | Status: new
Priority: major | Milestone: sage-3.4.2
Component: number theory | Keywords:
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The attached patch (relative to 3.4.1.rc4) implements a number of changes
to the number field
code. All have corresponding doctests verifying the new behaviour.
In particular:
* The following now works:
{{{
sage: K.<a> = NumberField(x^2 + 5)
sage: L.<b> = K.extension(x^2 + 1)
sage: L.ideal(K.ideal(2, a + 1))
Fractional ideal (-b - 1)
}}}
* `K.prime_factors` is defined for a number field `K`. Thus
{{{
sage: CyclotomicField(3).prime_factors(7)
[Fractional ideal (-2 *zeta3 + 1), Fractional ideal (2 *zeta3 + 3)]
}}}
* `K.pari_polynomial('a')` is now valid when `K` is a relative number
field.
Previously one could only specify the variable name for an absolute
number field.
* `list` has been rewritten for the class `CyclotomicFieldHomset` in
`morphism.py`.
Previously this failed for homomorphisms from `CyclotomicField(n)` to a
non-cyclotomic number_field if there were no `n`-th roots of unity in
the
codomain. The new code uses `zeta_order` to check whether there are
any
homomorphisms. This should be faster.
* In `number_field_ideal.py`, `element_1_mod` has been rewritten to use
the
pari function `idealaddtoone0`. This is considerably faster.
* `maximal_order` has been rewritten for relative number fields. The
previous version was repetitive and grossly wasteful of memory. This
solves one aspect of #4738 ("finding maximal_orders of relative number
fields is very slow").
* `is_principal` and `gens_reduced` have been changed in
`number_field_ideal_rel.py` so that if the ideal is principal, there is
only one reduced generator (as already happens for ideals in absolute
number
fields). The previous code did not work in all cases. As a result,
the output of several doctests have needed changing, and (mostly)
simplifying.
* Minor change have been made to `reduced_basis` so that it works for
cyclotomic
fields:
{{{
sage: CyclotomicField(12).reduced_basis()
[1, zeta12^2, zeta12, zeta12^3]
}}}
* `zeta_function` now works for relative number fields.
* In several places the code has been tidied up using the `map` function.
* A version of `uniformizer` has been introduced for relative number
fields.
* The valuation at a prime ideal for elements of relative number fields
has
been defined.
* A minor change to the `conjugate` function settles #4725.
* `relative_discriminant` in `number_field_rel.py` has been rewritten so
that it
uses PARI's `nf` rather than `bnf`, and is able to handle relative
fields whose
base field is a relative field.
* Made minor change to error messages for `zeta` in `number_field.py` and
`order.py` to make them compatible.
* A `subfields` function for relative number fields has been introduced.
Doctests have been added, and a minor change made, to the absolute
version.
* The relative version of `automorphisms` has been rewritten so that it
handles
correctly fields where the base field is relative.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5842>
Sage <http://sagemath.org/>
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