Function: rnfidealreltoabs
Section: number_fields
C-Name: rnfidealreltoabs0
Prototype: GGD0,L,
Help: rnfidealreltoabs(rnf,x,{flag=0}): transforms the ideal x from relative to
 absolute representation. As a vector of t_POLMODs if flag = 0 and as an ideal
 in HNF in the absolute field if flag = 1.
Doc: Let $\var{rnf}$ be a relative
 number field extension $L/K$ as output by \kbd{rnfinit} and let $x$ be a
 relative ideal, given as a $\Z_K$-module by a pseudo matrix $[A,I]$.
 This function returns the ideal $x$ as an absolute ideal of $L/\Q$.
 If $\fl = 0$, the result is given by a vector of \typ{POLMOD}s modulo
 \kbd{rnf.pol} forming a $\Z$-basis; if $\fl = 1$, it is given in HNF in terms
 of the fixed $\Z$-basis for $\Z_L$, see \secref{se:rnfinit}.
 \bprog
 ? K = nfinit(y^2+1); rnf = rnfinit(K, x^2-y);
 ? P = idealprimedec(K,2)[1];
 ? P = rnfidealup(rnf, P)
 %3 = [2, x^2 + 1, 2*x, x^3 + x]
 ? Prel = rnfidealhnf(rnf, P)
 %4 = [[1, 0; 0, 1], [[2, 1; 0, 1], [2, 1; 0, 1]]]
 ? rnfidealreltoabs(rnf,Prel)
 %5 = [2, x^2 + 1, 2*x, x^3 + x]
 ? rnfidealreltoabs(rnf,Prel,1)
 %6 =
 [2 1 0 0]

 [0 1 0 0]

 [0 0 2 1]

 [0 0 0 1]
 @eprog
 The reason why we do not return by default ($\fl = 0$) the customary HNF in
 terms of a fixed $\Z$-basis for $\Z_L$ is precisely because
 a \var{rnf} does not contain such a basis by default. Completing the
 structure so that it contains a \var{nf} structure for $L$ is polynomial
 time but costly when the absolute degree is large, thus it is not done by
 default. Note that setting $\fl = 1$ will complete the \var{rnf}.
Variant: Also available is
 \fun{GEN}{rnfidealreltoabs}{GEN rnf, GEN x} ($\fl = 0$).