Function: nfdisc
Section: number_fields
C-Name: nfdisc
Prototype: G
Help: nfdisc(T): discriminant of the number field defined by
the polynomial T. An argument [T,listP] is possible, where listP is a list
of primes or a prime bound.
Doc: \idx{field discriminant} of the number field defined by the integral,
preferably monic, irreducible polynomial $T(X)$. Returns the discriminant of
the number field $\Q[X]/(T)$, using the Round $4$ algorithm.
\misctitle{Local discriminants, valuations at certain primes}
As in \kbd{nfbasis}, the argument $T$ can be replaced by $[T,\var{listP}]$,
where \kbd{listP} is as in \kbd{nfbasis}: a vector of
pairwise coprime integers (usually distinct primes), a factorization matrix,
or a single integer. In that case, the function returns the discriminant of
an order whose basis is given by \kbd{nfbasis(T,listP)}, which need not be
the maximal order, and whose valuation at a prime entry in \kbd{listP} is the
same as the valuation of the field discriminant.
In particular, if \kbd{listP} is $[p]$ for a prime $p$, we can
return the $p$-adic discriminant of the maximal order of $\Z_p[X]/(T)$,
as a power of $p$, as follows:
\bprog
? padicdisc(T,p) = p^valuation(nfdisc([T,[p]]), p);
? nfdisc(x^2 + 6)
%2 = -24
? padicdisc(x^2 + 6, 2)
%3 = 8
? padicdisc(x^2 + 6, 3)
%4 = 3
@eprog\noindent The following function computes the discriminant of the
maximal order under the assumption that $P$ is a vector of prime numbers
containing (at least) all prime divisors of the field discriminant:
\bprog
globaldisc(T, P) =
{ my (D = nfdisc([T, P]));
sign(D) * factorback(P, [valuation(D,p) | p <-P]);
}
? globaldisc(x^2 + 6, [2, 3, 5])
%1 = -24
@eprog
\synt{nfdisc}{GEN T} (\kbd{listP = NULL}). Also available is
\fun{GEN}{nfbasis}{GEN T, GEN *d, GEN listP = NULL}, which returns the order
basis, and where \kbd{*d} receives the order discriminant.