1Function: nfdisc 2Section: number_fields 3C-Name: nfdisc 4Prototype: G 5Help: nfdisc(T): discriminant of the number field defined by 6 the polynomial T. An argument [T,listP] is possible, where listP is a list 7 of primes or a prime bound. 8Doc: \idx{field discriminant} of the number field defined by the integral, 9 preferably monic, irreducible polynomial $T(X)$. Returns the discriminant of 10 the number field $\Q[X]/(T)$, using the Round $4$ algorithm. 11 12 \misctitle{Local discriminants, valuations at certain primes} 13 14 As in \kbd{nfbasis}, the argument $T$ can be replaced by $[T,\var{listP}]$, 15 where \kbd{listP} is as in \kbd{nfbasis}: a vector of pairwise coprime 16 integers (usually distinct primes), a factorization matrix, or a single 17 integer. In that case, the function returns the discriminant of an order 18 whose basis is given by \kbd{nfbasis(T,listP)}, which need not be the maximal 19 order, and whose valuation at a prime entry in \kbd{listP} is the same as the 20 valuation of the field discriminant. 21 22 In particular, if \kbd{listP} is $[p]$ for a prime $p$, we can 23 return the $p$-adic discriminant of the maximal order of $\Z_p[X]/(T)$, 24 as a power of $p$, as follows: 25 \bprog 26 ? padicdisc(T,p) = p^valuation(nfdisc([T,[p]]), p); 27 ? nfdisc(x^2 + 6) 28 %2 = -24 29 ? padicdisc(x^2 + 6, 2) 30 %3 = 8 31 ? padicdisc(x^2 + 6, 3) 32 %4 = 3 33 @eprog\noindent The following function computes the discriminant of the 34 maximal order under the assumption that $P$ is a vector of prime numbers 35 containing (at least) all prime divisors of the field discriminant: 36 \bprog 37 globaldisc(T, P) = 38 { my (D = nfdisc([T, P])); 39 sign(D) * vecprod([p^valuation(D,p) | p <-P]); 40 } 41 ? globaldisc(x^2 + 6, [2, 3, 5]) 42 %1 = -24 43 @eprog 44 45 \synt{nfdisc}{GEN T}. Also available is \fun{GEN}{nfbasis}{GEN T, GEN *d}, 46 which returns the order basis, and where \kbd{*d} receives the order 47 discriminant. 48