\\ 2: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ T = x^6 + 2854*x^4 + 2036329*x^2 + 513996528; K = bnfinit(T); K.fu \\ missing units K = bnfinit(T, 1); \\ impose units computation K.fu 2^2^100 \\ 3: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ K = nfinit(x^2 - 2); K.pol K.zk K.sign K.r1 K.r2 K.disc K.clgp \\ fails K.fu \\ fails K = bnfinit(x^2 - 2); \\ or K = bnfinit(K) K.clgp K.fu \\ 4: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ K = nfinit(x^3 - 2); nfeltmul(K, x, x^2+1) nfelttrace(K, x+1) nfeltadd(K, x/2, [1,2,3]~) nfbasistoalg(K, %) nfalgtobasis(K, %) \\ 6: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ 2^1000 * 3^-2000 \\ 9: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ D = 1000001273; K = bnfinit(x^2 - D, 1); bnfunits(K) K.fu P = idealprimedec(K,2)[1]; bnfisprincipal(K, P) bnfisprincipal(K, P, 4) \\ factored representation bnfisprincipal(K, P, 3) \\ expanded; no longer do this ! \\10: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ S = idealprimedec(K,2); U = bnfunits(K, S) bnfisunit(K, 2) \\ not a unit bnfisunit(K, 2, U) \\ ... but an S-unit \\11ff: \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ K = nfinit(x^3 - 2); u = [x, 2; [1,2,3]~,-1] nffactorback(K, u) v = [x+1, 1; [-1,2,3]~,2] nfeltmul(K, u, v) nfeltpow(K, u, 2) nfeltdiv(K, u, 2) nfeltnorm(K, u) nffactorback(K, [u,v], [2,3]) \\ still factored nffactorback(K, %) \\ now expand completely nfelttrace(K, u) \\ not multiplicative ! P = idealprimedec(K,5)[2]; nfmodpr(K, v, P) bid = idealstar(K, 5); ideallog(K, v, bid) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\