Karim Belabas on Thu, 24 Jan 2013 16:11:06 +0100

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 Re: Field inclusion problem

```* Ewan Delanoy [2013-01-24 15:56]:
> >The problem is variable priorities. The variable of the polynomial
>  >defining the "base field" must have *lower* priority then the variable
>  >of the polynomial to be factored. See ??nffactor
>
>
>  The output of ??nffactor contains the following : â(see Section [Label: se:priority])â. What does that refer to ?
>  Is it a section in the userâs manual or in the online help ? I tried ??Label but to no avail.

The online help is directly extracted from the (TeX) documentation
(after a rough formatting attempt). Unfortunately it can't follow
cross-references.

I recommand to read the actual pdf documentation, in the GP User's Manual,
Chapter 2, section "Variable priorities, multivariate objects".

It's also possible to search a little through the online help:

(16:05) gp > ???"se:priority"@          \\ @ means : include all chapters

Chapter 3:
==========
denominator      factornf         galoisfixedfield nffactor
nffactormod      nfroots          numerator        rnfdisc
rnfequation      rnfinit          writebin

Relative extensions

Chapter 4:
==========
Multivariate objects
Type \typ{POLMOD} (polmod)
Type \typ{POL} (polynomial)

Chapter 2:
==========
Polmods (\typ{POLMOD})
Variable priorities, multivariate objects

This one is the most interesting:

(16:05) gp > ?? "Variable priorities, multivariate objects"@
Variable priorities, multivariate objects:

A  multivariate  polynomial in PARI is just a polynomial  (in one variable),
whose coefficients are themselves polynomials,  arbitrary but for the fact that
they do not involve the main variable. (PARI currently has no sparse
representation for polynomials, listing only non-zero monomials.) All
computations  are  then  done formally on the coefficients as if the polynomial
was univariate.

This  is  not  symmetrical.    So if I enter x + y in a clean session,  what
happens?   This is understood as  x^1 + (y^1 + 0*y^0)*x^0 belongs to (Z[y])[x]
but  how  do we know that x is "more important" than y ?   Why not y^1 + x*y^0,
which is the same mathematical entity after all?

[ ...etc...]

(But if you never went through GP's documentation, it's easier to read a PDF.)

Cheers,

K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite Bordeaux 1          Fax: (+33) (0)5 40 00 69 50
351, cours de la Liberation    http://www.math.u-bordeaux1.fr/~belabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux1.fr/  [PARI/GP]
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