f = x^4 - 2*x^3 + x^2 - 5;
polisirreducible(f)

g = polcyclo(30)

Mod(x,f)^5

lift(Mod(x,g)^15)

{h = x^5 + 7*x^4 + 22550*x^3 - 281686*x^2 - 85911*x + 3821551};
polredbest(h)

K = nfinit(f);

K.pol
K.sign
K.disc
K.zk
w = K.zk[2];

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]]

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

bnfcertify(K2)

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)

bnf = bnfinit(y^2-y+50);
bnf.cyc
R = bnrclassfield(bnf)[1]
[cond,bnr,subg] = rnfconductor(bnf,R);
cond
subg
R2 = bnrclassfield(bnf,,2)

bnr = bnrinit(bnf,12);
bnr.cyc
[deg,r1,D] = bnrdisc(bnr);
[deg,r1]
D
bnrclassfield(bnr)
bnrclassfield(bnr,,1)

bnr = bnrinit(bnf,7);
bnr.cyc
bnrclassfield(bnr,3)
pr41 = idealprimedec(bnf,41)[1];
bnrisprincipal(bnr,pr41,0)

bnr = bnrinit(bnf,[102709,43512;0,1],1);
bnr.cyc
bnrclassfield(bnr,[9,3;0,1])

bnf=bnfinit(y^2-217);
bnf.cyc
bnrinit(bnf,1).cyc
bnrinit(bnf,[1,[1,1]]).cyc