Function: rnfpolredabs
Section: number_fields
C-Name: rnfpolredabs
Prototype: GGD0,L,
Help: rnfpolredabs(nf,pol,{flag=0}): given an irreducible pol with coefficients
 in nf, finds a canonical relative polynomial defining the same field.
 Binary digits of flag mean: 1: return also the element whose characteristic
 polynomial is the given polynomial, 2: return an absolute polynomial,
 16: partial reduction.
Doc: Relative version of \kbd{polredabs}. Given an irreducible monic polynomial
 \var{pol} with coefficients in the maximal order of $\var{nf}$, finds a
 canonical relative
 polynomial defining the same field, hopefully with small coefficients.
 Note that the equation is only canonical for a fixed \var{nf}, using a
 different defining polynomial in the \var{nf} structure will produce a
 different relative equation.

 The binary digits of $\fl$ correspond to $1$: add information to convert
 elements to the new representation, $2$: absolute polynomial, instead of
 relative, $16$: possibly use a suborder of the maximal order. More precisely:

 0: default, return $P$

 1: returns $[P,a]$ where $P$ is the default output and $a$,
 a \typ{POLMOD} modulo $P$, is a root of \var{pol}.

 2: returns \var{Pabs}, an absolute, instead of a relative, polynomial.
 This polynomial is canonical and does not depend on the \var{nf} structure.
 Same as but faster than
 \bprog
   polredabs(rnfequation(nf, pol))
 @eprog

 3: returns $[\var{Pabs},a,b]$, where \var{Pabs} is an absolute polynomial
 as above, $a$, $b$ are \typ{POLMOD} modulo \var{Pabs}, roots of \kbd{nf.pol}
 and \var{pol} respectively.

 16: (OBSOLETE) possibly use a suborder of the maximal order. This makes
 \kbd{rnfpolredabs} behave as \kbd{rnfpolredbest}. Just use the latter.

 \misctitle{Warning} The complexity of \kbd{rnfpolredabs}
 is exponential in the absolute degree. The function \tet{rnfpolredbest} runs
 in polynomial time, and  tends  to return polynomials with smaller
 discriminants. It also supports polynomials with arbitrary coefficients in
 \var{nf}, neither integral nor necessarily monic.
