1Function: nfgrunwaldwang 2Section: number_fields 3C-Name: nfgrunwaldwang 4Prototype: GGGGDn 5Help: nfgrunwaldwang(nf,Lpr,Ld,pl,{v='x}): a polynomial in the variable v 6 defining a cyclic extension of nf (given in nf or bnf form) with local 7 behavior prescribed by Lpr, Ld and pl: the extension has local degree a 8 multiple of Ld[i] at the prime Lpr[i], and the extension is complex at the 9 i-th real place of nf if pl[i]=-1 (no condition if pl[i]=0). The extension 10 has degree the LCM of the local degrees. 11Doc: Given \var{nf} a number field in \var{nf} or \var{bnf} format, 12 a \typ{VEC} \var{Lpr} of primes of \var{nf} and a \typ{VEC} \var{Ld} of 13 positive integers of the same length, a \typ{VECSMALL} \var{pl} of length 14 $r_1$ the number of real places of \var{nf}, computes a polynomial with 15 coefficients in \var{nf} defining a cyclic extension of \var{nf} of 16 minimal degree satisfying certain local conditions: 17 18 \item at the prime~$Lpr[i]$, the extension has local degree a multiple 19 of~$Ld[i]$; 20 21 \item at the $i$-th real place of \var{nf}, it is complex if $pl[i]=-1$ 22 (no condition if $pl[i]=0$). 23 24 The extension has degree the LCM of the local degrees. Currently, the degree 25 is restricted to be a prime power for the search, and to be prime for the 26 construction because of the \kbd{rnfkummer} restrictions. 27 28 When \var{nf} is $\Q$, prime integers are accepted instead of \kbd{prid} 29 structures. However, their primality is not checked and the behavior is 30 undefined if you provide a composite number. 31 32 \misctitle{Warning} If the number field \var{nf} does not contain the $n$-th 33 roots of unity where $n$ is the degree of the extension to be computed, 34 the function triggers the computation of the \var{bnf} of $nf(\zeta_n)$, 35 which may be costly. 36 37 \bprog 38 ? nf = nfinit(y^2-5); 39 ? pr = idealprimedec(nf,13)[1]; 40 ? pol = nfgrunwaldwang(nf, [pr], [2], [0,-1], 'x) 41 %3 = x^2 + Mod(3/2*y + 13/2, y^2 - 5) 42 @eprog 43