Make number fields work when polynomial not integral or not monic.

[williamstein]

The goal of this ticket is to make number fields work when the defining polynomial is not integral or not monic.

If the user specifies a non-integral or non-monic polynomial to define an absolute or relative number field, we define the corresponding PARI number field by a monic integral polynomial obtained from the PARI functions polredbest and rnfpolredbest, respectively.

The new methods NumberField_generic._pari_absolute_structure() and NumberField_relative._pari_relative_structure() return the data needed to convert elements between the Sage NumberField and the PARI nf structure.

Depends on #18740

CC: @katestange @sagetrac-Bouillaguet

Component: number fields

Author: Peter Bruin

Branch/Commit: 8b86c52

Reviewer: Kartik Venkatram

Issue created by migration from https://trac.sagemath.org/ticket/252

[sagetrac-cwitty]

comment:2

You may find sage.rings.algebraic_real.clear_denominators() useful here. (If so, the function should probably be moved to a more sensible place, and perhaps renamed.)

[williamstein]

comment:3

The example above works. But other things don't:

sage: R.<x> = QQ[]
sage: sage: L.<b> = NumberField(x^2-1/2)
sage: sage: L.discriminant()
8
sage: L.ring_of_integers()
boom

[sagetrac-mabshoff]

comment:4

Notice that #4041 is a duplicate of this ticket.

Cheers,

Michael

[lftabera]

comment:6

Another example from #9408

sage: L.<a,b> = QQ[i].relativize(1) #Ok
sage: L.<a,b> = QQ[i].relativize(1/2) #PariError