Function: idealprimedec
Section: number_fields
C-Name: idealprimedec_limit_f
Prototype: GGD0,L,
Description:
(gen, gen):vec idealprimedec($1, $2)
(gen, gen, ?small):vec idealprimedec_limit_f($1, $2, $3)
Help: idealprimedec(nf,p,{f=0}): prime ideal decomposition of the prime number
p in the number field nf as a vector of prime ideals. If f is present
and non-zero, restrict the result to primes of residue degree <= f.
Doc: computes the prime ideal
decomposition of the (positive) prime number $p$ in the number field $K$
represented by \var{nf}. If a non-prime $p$ is given the result is undefined.
If $f$ is present and non-zero, restrict the result to primes of residue
degree $\leq f$.
The result is a vector of \tev{prid} structures, each representing one of the
prime ideals above $p$ in the number field $\var{nf}$. The representation
$\kbd{pr}=[p,a,e,f,\var{mb}]$ of a prime ideal means the following: $a$
is an algebraic integer in the maximal order $\Z_K$ and the prime ideal is
equal to $\goth{p} = p\Z_K + a\Z_K$;
$e$ is the ramification index; $f$ is the residual index;
finally, \var{mb} is the multiplication table attached to the algebraic
integer $b$ is such that $\goth{p}^{-1}=\Z_K+ b/ p\Z_K$, which is used
internally to compute valuations. In other words if $p$ is inert,
then \var{mb} is the integer $1$, and otherwise it is a square \typ{MAT}
whose $j$-th column is $b \cdot \kbd{nf.zk[j]}$.
The algebraic number $a$ is guaranteed to have a
valuation equal to 1 at the prime ideal (this is automatic if $e>1$).
The components of \kbd{pr} should be accessed by member functions: \kbd{pr.p},
\kbd{pr.e}, \kbd{pr.f}, and \kbd{pr.gen} (returns the vector $[p,a]$):
\bprog
? K = nfinit(x^3-2);
? P = idealprimedec(K, 5);
? #P \\ 2 primes above 5 in Q(2^(1/3))
%3 = 2
? [p1,p2] = P;
? [p1.e, p1.f] \\ the first is unramified of degree 1
%5 = [1, 1]
? [p2.e, p2.f] \\ the second is unramified of degree 2
%6 = [1, 2]
? p1.gen
%7 = [5, [2, 1, 0]~]
? nfbasistoalg(K, %[2]) \\ a uniformizer for p1
%8 = Mod(x + 2, x^3 - 2)
? #idealprimedec(K, 5, 1) \\ restrict to f = 1
%9 = 1 \\ now only p1
@eprog