Function: nfisisom
Section: number_fields
C-Name: nfisisom
Prototype: GG
Help: nfisisom(f,g): as nfisincl but tests whether f is isomorphic to g.
Doc: as \tet{nfisincl}, but tests for isomorphism. More efficient if
$f$ or $g$ is a number field structure.
\bprog
? f = x^6 + 30*x^5 + 495*x^4 + 1870*x^3 + 16317*x^2 - 22560*x + 59648;
? g = x^6 + 42*x^5 + 999*x^4 + 8966*x^3 + 36117*x^2 + 21768*x + 159332;
? h = x^6 + 30*x^5 + 351*x^4 + 2240*x^3 + 10311*x^2 + 35466*x + 58321;
? #nfisisom(f,g) \\ two isomorphisms
%3 = 2
? nfisisom(f,h) \\ not isomorphic
%4 = 0
\\ comparative bench
? K = nfinit(f); L = nfinit(g); B = 10^3;
? for (i=1, B, nfisisom(f,g))
time = 6,124 ms.
? for (i=1, B, nfisisom(K,g))
time = 3,356 ms.
? for (i=1, B, nfisisom(f,L))
time = 3,204 ms.
? for (i=1, B, nfisisom(K,L))
time = 3,173 ms.
@eprog\noindent
The function is usually very fast when the fields are non-isomorphic,
whenever the fields can be distinguished via a simple invariant such as
degree, signature or discriminant. It may be slower when the fields
share all invariants, but still faster than computing actual isomorphisms:
\bprog
\\ usually very fast when the answer is 'no':
? for (i=1, B, nfisisom(f,h))
time = 32 ms.
\\ but not always
? u = x^6 + 12*x^5 + 6*x^4 - 377*x^3 - 714*x^2 + 5304*x + 15379
? v = x^6 + 12*x^5 + 60*x^4 + 166*x^3 + 708*x^2 + 6600*x + 23353
? nfisisom(u,v)
%13 = 0
? polsturm(u) == polsturm(v)
%14 = 1
? nfdisc(u) == nfdisc(v)
%15 = 1
? for(i=1,B, nfisisom(u,v))
time = 1,821 ms.
? K = nfinit(u); L = nfinit(v);
? for(i=1,B, nfisisom(K,v))
time = 232 ms.
@eprog