Abstract

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.

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Mathematical Subject Classification 2010

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