default(new_galois_format,1);
P1 = x^4-5;
polgalois(P1)

P2 = x^4-x^3-7*x^2+2*x+9;
polgalois(P2)

P3 = x^4-x^3-3*x^2+x-1;
polgalois(P3)

nflist([4,3],2000)
nflist([4,4],[1,10000])
nflist([4,5],[1,5000],0)

Q1 = nfsplitting(P1)
Q2 = nfsplitting(P2)

Q3 = nfsplitting(P3)

Q3 = polredbest(Q3)

gal = galoisinit(Q3);
gal.gen

ord = gal.orders
prod(i=1,#ord,ord[i])

galoisidentify(gal)
galoisgetname(24,12)

L = galoissubgroups(gal);
#L

R1 = galoisfixedfield(gal,L[25])[1];
polgalois(R1)
R2 = galoisfixedfield(gal,L[28])[1];
polgalois(R2)

nf = nfinit(Q3);
factor(nf.disc)

dec3 = idealprimedec(nf,3);
pr3 = dec3[1];
[#dec3, pr3.f, pr3.e]

ram3 = idealramgroups(nf,gal,pr3);
#ram3

galoisidentify(ram3[1])
galoisisabelian(ram3[1])

galoisidentify(ram3[2])

galoisidentify(ram3[3])

dec11 = idealprimedec(nf,11);
pr11 = dec11[1];
[#dec11, pr11.f, pr11.e]

ram11 = idealramgroups(nf,gal,pr11);
#ram11

galoisidentify(ram11[1])
galoisidentify(ram11[2])

dec2 = idealprimedec(nf,2);
pr2 = dec2[1];
[#dec2, pr2.f, pr2.e]
frob2 = idealfrobenius(nf,gal,pr2);
permorder(frob2)

N = 7*13*19;
L1 = polsubcyclo(N,3);

L2 = [P | P <- L1, #factor(nfinit(P).disc)[,1]==3]

G = znstar(N)

H = mathnfmodid([1,0;-1,1;0,-1],3);

pol = galoissubcyclo(G,H)
factor(nfinit(pol).disc)

bnf = bnfinit(a^2-a+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) \\3-sous-ext. élémentaire

pr41 = idealprimedec(bnf,41)[1];
bnrisprincipal(bnr,pr41,0)

bnr = bnrinit(bnf,[102709,43512;0,1]);
bnr.cyc
M = [9,3;0,1]; \\sous-groupe d'indice 9
bnrclassfield(bnr,M)

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

bnf = bnfinit(a^2+23);
bnr = bnrinit(bnf,1);
bnr.cyc

r = lfun([bnr,[1]],0,1)

R2 = algdep(exp(r),3)
nfdisc(R2)

quadhilbert(-31)
lift(quadray(13,7))

bnf = bnfinit(x^2+2*3*5*7*11);
bnf.cyc
bnr = bnrinit(bnf,1,1);
gal = galoisinit(bnf);
m = bnrgaloismatrix(bnr,gal)[1]

bnf=bnfinit(a^2+47);
bnf.cyc
P=quadhilbert(-47)
[A,J] = rnfpseudobasis(bnf,P);
A
J

[As,Js] = rnfsteinitz(bnf,[A,J]);
Js
bnfisprincipal(bnf, Js[#Js])