Function: polsubcyclofast
Section: polynomials
C-Name: polsubcyclofast
Prototype: GLD0,L,D0,L,
Help: polsubcyclofast(n,d,{s=0},{exact=0}): If 1 <= d <= 6
 or a prime, finds an equation for the subfields of Q(zeta_n) with galois
 group C_d. Contrary to polsubcyclo, the output is
 always a (possibly empty) vector of polynomials. If s = 0 (default) all
 signatures, otherwise s = 1 (resp., -1) for totally real (resp., totally
 complex). Set exact = 1 for subfields of exact conductor n.
Doc: If $1 \leq d\leq 6$ or a prime, finds an equation for the subfields of
 $\Q(\zeta_{n})$ with Galois group $C_{d}$; the special value $d = -4$ provides
 the subfields with group $V_{4}=C_{2}\times C_{2}$. Contrary to
 \kbd{polsubcyclo}, the
 output is always a (possibly empty) vector of polynomials. If $s = 0$ (default)
 all signatures, otherwise $s = 1$ (resp., $-1$) for totally real
 (resp., totally complex). Set \kbd{exact = 1} for subfields of conductor $n$.

 The argument $n$ can be given as in arithmetic functions: as an integer, as a
 factorization matrix, or (preferred) as a pair $[N, \kbd{factor}(N)]$.

 \misctitle{Comparison with \kbd{polsubcyclo}} First \kbd{polsubcyclofast}
 does not usually return Gaussian periods, but ad hoc polynomials which do
 generate the same field. Roughly speaking (ignoring
 logarithmic factors), the complexity of \kbd{polsubcyclo} is independent of
 $d$ and the complexity of \kbd{polsubcyclofast} is independent of $n$.
 Ignoring logarithmic factors, \kbd{polsubcylo} runs in time $O(n)$ and
 \kbd{polsubcyclofast} in time $O(d^{4})$.
 So the latter is \emph{much} faster than \kbd{polsubcyclo} if $n$ is large,
 but gets slower as $d$ increases and becomes unusable for $d \geq 40$ or so.

 \bprog
 ? polsubcyclo(10^7+19,7);
 time = 1,852 ms.
 ? polsubcyclofast(10^7+19,7);
 time = 15 ms.

 ? polsubcyclo(10^17+21,5); \\ won't finish
  *** polsubcyclo: user interrupt after 2h
 ? polsubcyclofast(10^17+21,5);
 time = 3 ms.

 ? polsubcyclofast(10^17+3,7);
 time = 26 ms.

 ? polsubcyclo(10^6+117,13);
 time = 193 ms.
 ? polsubcyclofast(10^6+117,13);
 time = 50 ms.

 ? polsubcyclofast(10^6+199,19);
 time = 202 ms.
 ? polsubcyclo(10^6+199,19); \\ about as fast
 time = 3191ms.

 ? polsubcyclo(10^7+271,19);
 time = 2,067 ms.
 ? polsubcyclofast(10^7+271,19);
 time = 201 ms.
 @eprog
