Function: nfsubfields
Section: number_fields
C-Name: nfsubfields0
Prototype: GD0,L,D0,L,
Help: nfsubfields(pol,{d=0},{flag=0}): finds all subfields of degree d of number
 field defined by pol (all subfields if d is null or omitted). The result is a
 vector of subfields, each being given by [g,h] (default) or simply g (flag=1),
 where g is an absolute equation and h expresses one of the roots of g in terms
 of the root x of the polynomial defining nf.
Doc: finds all subfields of degree
 $d$ of the number field defined by the (monic, integral) polynomial
 \var{pol} (all subfields if $d$ is null or omitted). The result is a vector
 of subfields, each being given by $[g,h]$ (default) or simply $g$ ($\fl=1$),
 where $g$ is an absolute equation
 and $h$ expresses one of the roots of $g$ in terms of the root $x$ of the
 polynomial defining $\var{nf}$. This routine uses

 \item Allombert's \tet{galoissubfields} when \var{nf} is Galois (with weakly
 supersolvable Galois group).\sidx{Galois}\sidx{subfield}

 \item Kl\"uners's or van Hoeij--Kl\"uners--Novocin algorithm
 in the general case. The latter runs in polynomial time and is generally
 superior unless there exists a small unramified prime $p$ such that \var{pol}
 has few irreducible factors modulo $p$.

 An input of the form~\kbd{[nf, fa]} is also allowed, where~\kbd{fa} is the
 factorisation of~\var{nf.pol} over~\var{nf}, expressed as a famat of
 polynomials with coefficients in the variable of~\kbd{nf}, in which case the
 van Hoeij--Kl\"uners--Novocin algorithm is used.

 \bprog
 ? pol = x^4 - x^3 - x^2 + x + 1;
 ? nfsubfields(pol)
 %2 = [[x, 0], [x^2 - x + 1, x^3 - x^2 + 1], [x^4 - x^3 - x^2 + x + 1, x]]
 ? nfsubfields(pol,,1)
 %2 = [x, x^2 - x + 1, x^4 - x^3 - x^2 + x + 1]
 ? y=varhigher("y"); fa = nffactor(pol,subst(pol,x,y));
 ? #nfsubfields([pol,fa])
 %5 = 3
 @eprog

Variant: Also available is \fun{GEN}{nfsubfields}{GEN nf, long d}, corresponding
 to $\fl = 0$.
