Orders in number fields

[Justin C. Walker]

Hi, all,

I stumbled across what seems to be a bug/missing feature/misunderstanding while experimenting with orders.

Let f be a univariate polynomial (say, over QQ). I used both quadratic and cubic polynomials.

Then the following leads to an unexpected problem:

K.<a> = NumberField(f)
OK = K.maximal_order()
O1 = K.order(a)
O2 = ZZ[a]

Then I get

O1.ambient() is O2.ambient()
False
(the two are isomorphic).

This causes problems, for example, in

O2.index_in(OK)

because “self” and “other” have different ambient fields.

I tried to figure out why the “ZZ[a]” construction doesn’t work as I would expect, but got lost in a maze of calls and caches :-}

As a side issue, O2 does not have a coercion map to OK. Is that intended?

Thoughts, pointers, etc. always appreciated.

Thanks, in advance.

Justin

--
Justin C. Walker, Curmudgeon at Large
Institute for the Absorption of Federal Funds
-----------
I want to die, peacefully in my sleep, like my grandfather;
not screaming in terror, like his passengers.

[Samuel Lelievre]

Orders in number fields certainly need work.

This Sage Trac query:

    https://trac.sagemath.org/query?order=id&desc=1&summary=~orders

and similar queries with "order" or "ring of integers" reveal the following open tickets

    #28706 clarify order generation in number fields with bracket syntax

    #28552  Elements in orders in number fields are considered as field elements

    #27413  Implement picard group and unit group for non-maximal orders

    #25709  inject_variables is broken for orders of number fields

    #24934  Orders are not unique parents

    #24031  Coercion between Matrices over orders and over the number field

    #23971  Euclidean division in quadratic imaginary orders

    #23321  Implemented functionality for Quotient Rings of Orders

    #23236  Create p-maximal orders

    #18865  Can't make ring homomorphism from ring of integers to a residue field

    #16556  AbsoluteOrder.random_element and PolynomialQuotientRing_integer.random_element

    #14740  Infinite loop in creation of number field order

    #13568  Quaternion Orders over Number Fields

    #12242  Divisibility of number field order elements needs work

    #5893   Norm Form for Number Fields and Orders

    #4738   base_ring of orders in relative number fields is wrong

    #1134   optimize creating elements of orders and number fields by coercing in lists

I can also link to discussions on other channels if that seems useful.

[John Cremona]

I'll forward this to sage-nt. I have not used orders in number fields
much myself, but it looks to me as if several of these should be easy.
One of them (about quaternion orders) is altogether different since
that is about orders in quaternion algebras (over number fields), not
about orders in number fields.

John

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