Let
$L=\mathbb{Q}\left(\sqrt{{d}_{1}},\dots ,\sqrt{{d}_{n}}\right)$ be a real
multiquadratic field and
$S$
be a set of prime ideals of
$L$.
In this paper, we present a heuristic algorithm for the computation of the
$S$-class group and
the
$S$-unit group
that runs in time
$Poly\left(log\left(\Delta \right),Size\left(S\right)\right){e}^{\tilde{O}\left(\sqrt{lnd}\right)}$
where
$d={\prod}_{i\le n}{d}_{i}$ and
$\Delta $ is the
discriminant of
$L$.
We use this method to compute the ideal class group of the maximal order
${\mathcal{O}}_{L}$ of
$L$ in time
$Poly\left(log\left(\Delta \right)\right){e}^{\tilde{O}\left(\sqrt{logd}\right)}$. When
$log\left(d\right)\le log{\left(log\left(\Delta \right)\right)}^{c}$ for some
constant
$c<2$,
these methods run in polynomial time. We implemented our algorithm using Sage
7.5.1.

Keywords

ideal class group, $S$-unit group, multiquadratic fields