f = x^4 - 2*x^3 + x^2 - 5;
K = nfinit(f)
#nfisisom(nfinit(P), nfinit(polredbest(P)))
K.pol
K.sign
K.r1
K.r2

K.disc
K.p
w=K.zk[2];
K.zk

nfalgtobasis(K,x^2)
nfbasistoalg(K,[1,1,1,1]~)

nfeltmul(K,[1,-1,0,0]~,x^2)
nfeltnorm(K,x-2)
nfelttrace(K,[0,1,2,0]~)

dec = idealprimedec(K,5);
#dec
[pr1,pr2] = dec;
pr1.f
pr1.e

pr1.gen
pr2.f
pr2.e

idealhnf(K,pr1)
a = idealhnf(K,[23, 10, -5, 1]~)
idealnorm(K,a)

idealpow(K,pr2,3)
idealnorm(K,idealadd(K,a,pr2))
fa = idealfactor(K,a);
matsize(fa)
[fa[1,1].p, fa[1,1].f, fa[1,1].e, fa[1,2]]
[fa[2,1].p, fa[2,1].f, fa[2,1].e, fa[2,2]]
fa[2,1]==pr1
[fa[3,1].p, fa[3,1].f, fa[3,1].e, fa[3,2]]
b = idealchinese(K,[pr1,2;pr2,1],[1,-1]);
nfeltval(K,b-1,pr1)
nfeltval(K,b+1,pr2)

K2 = bnfinit(K);
K2.nf == K
K2.no
K2.reg
lift(K2.tu)
K2.tu[1]==nfrootsof1(K)[1]
lift(K2.fu)

L = bnfinit(x^3 - x^2 - 54*x + 169);
L.cyc
L.gen
pr = idealprimedec(L,13)[1]
[dl,g] = bnfisprincipal(L,pr);
dl
g
{idealhnf(L,pr) == idealmul(L,g,idealfactorback(L,L.gen,dl))}
[dl2,g2] = bnfisprincipal(L,idealpow(L,pr,2));
dl2
g2
idealhnf(L,g2) == idealpow(L,pr,2)

bnfisintnorm(L,5)
bnfisintnorm(L,65)

u = [0,2,1]~;
nfeltnorm(L,u)
bnfisunit(L,u)
lift(L.fu)
lift(L.tu)