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              <p>In this paper we make a series of  numerical experiments to support  Greenberg's p-rationality conjecture, we present a family of p-rational biquadratic fields and we find new examples of p-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the p-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.</p>
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              <p>Dans cet article nous apportons des éléments en faveur de la conjecture de Greenberg d'existence de corps p-rationnels à groupe de Galois connu. Nous introduisons une famille de corps biquadratiques p-rationnels et nous donnons des nouveaux exemples numériques de corps p-rationnels multiquadratiques de grand degré. Dans le cas des corps multiquadratiques et multicubiques on prouve que la conjecture est une conséquence de la conjonction de l'heuristique de Cohen-Lenstra-Martinet et d'une conjecture de Hofmann et Zhang portant sur le régulateur p-adique; nous apportons des nouveaux résultats numériques en faveur de ces conjectures. Une comparaison des outils existants nous amène à proposer des modifiquations algorithmiques.</p>
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