Function: nfsnf
Section: number_fields
C-Name: nfsnf
Prototype: GG
Help: nfsnf(nf,x): if x=[A,I,J], outputs [c_1,...c_n] Smith normal form of x.
Doc: given a $\Z_K$-module $x$ associated to the integral pseudo-matrix
 $(A,I,J)$, returns an ideal list $d_1,\dots,d_n$ which is the \idx{Smith
 normal form} of $x$. In other words, $x$ is isomorphic to
 $\Z_K/d_1\oplus\cdots\oplus\Z_K/d_n$ and $d_i$ divides $d_{i-1}$ for $i\ge2$.

 See \secref{se:ZKmodules} for the definition of integral pseudo-matrix;
 briefly, it is input as a 3-component row vector $[A,I,J]$ where
 $I = [b_1,\dots,b_n]$ and $J = [a_1,\dots,a_n]$ are two ideal lists,
 and $A$ is a square $n\times n$ matrix with columns $(A_1,\dots,A_n)$,
 seen as elements in $K^n$ (with canonical basis $(e_1,\dots,e_n)$).
 This data defines the $\Z_K$ module $x$ given by
 $$ (b_1e_1\oplus\cdots\oplus b_ne_n) / (a_1A_1\oplus\cdots\oplus a_nA_n)
 \enspace, $$
 The integrality condition is $a_{i,j} \in b_i a_j^{-1}$ for all $i,j$. If it
 is not satisfied, then the $d_i$ will not be integral. Note that every
 finitely generated torsion module is isomorphic to a module of this form and
 even with $b_i=Z_K$ for all $i$.