Function: nfeltval
Section: number_fields
C-Name: gpnfvalrem
Prototype: GGGD&
Help: nfeltval(nf,x,pr,{&y}): valuation of element x at the prime pr as output
by idealprimedec.
Doc: given an element $x$ in
\var{nf} and a prime ideal \var{pr} in the format output by
\kbd{idealprimedec}, computes the valuation $v$ at \var{pr} of the
element $x$. The valuation of $0$ is \kbd{+oo}.
\bprog
? nf = nfinit(x^2 + 1);
? P = idealprimedec(nf, 2)[1];
? nfeltval(nf, x+1, P)
%3 = 1
@eprog\noindent
This particular valuation can also be obtained using
\kbd{idealval(\var{nf},x,\var{pr})}, since $x$ is then converted to a
principal ideal.
If the $y$ argument is present, sets $y = x \tau^v$, where $\tau$ is a
fixed ``anti-uniformizer'' for \var{pr}: its valuation at \var{pr} is $-1$;
its valuation is $0$ at other prime ideals dividing \kbd{\var{pr}.p} and
nonnegative at all other primes. In other words $y$ is the part of $x$
coprime to \var{pr}. If $x$ is an algebraic integer, so is $y$.
\bprog
? nfeltval(nf, x+1, P, &y); y
%4 = [0, 1]~
@eprog
For instance if $x = \prod_i x_i^{e_i}$ is known to be coprime to \var{pr},
where the $x_i$ are algebraic integers and $e_i\in\Z$ then,
if $v_i = \kbd{nfeltval}(\var{nf}, x_i, \var{pr}, \&y_i)$, we still
have $x = \prod_i y_i^{e_i}$, where the $y_i$ are still algebraic integers
but now all of them are coprime to \var{pr}. They can then be mapped to
the residue field of \var{pr} more efficiently than if the product had
been expanded beforehand: we can reduce mod \var{pr} after each ring
operation.
Variant: Also available are
\fun{long}{nfvalrem}{GEN nf, GEN x, GEN pr, GEN *y = NULL}, which returns
\tet{LONG_MAX} if $x = 0$ and the valuation as a \kbd{long} integer,
and \fun{long}{nfval}{GEN nf, GEN x, GEN pr}, which only returns the
valuation ($y = \kbd{NULL}$).