\\ 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)
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