부산대학교 도서관

The authors independently study the capitulation (i.e.\ the principalization) of ideal classes in cyclic unramified extensions of degree 2 or 3 over certain number fields of degree 2, 4 or 6. \par The first part of the paper is just a survey of A. Azizi's results on the capitulation of 2-subgroups of ideal classes in unramified quadratic extensions over biquadratic number fields $K = \Bbb Q [i, \sqrt d ]$ with 2-group of ideal classes isomorphic to $C_2 \times C_2$ [C. R. Acad. Sci. Paris Sér. I Math. {\bf 325} (1997), no.~2, 127--130; MR1467063 (98d:11131)]. \par The second part of the paper under review is devoted to the study of the capitulation of the 3-subgroup of ideal classes in unramified cyclic 3-extensions over absolute cubic cyclic number fields $K$ with 3-group of ideal classes isomorphic to $C_3 \times C_3$. It comes from the doctoral dissertation of M. Ayadi [``Sur la capitulation des 3-classes d'idéaux d'un corps cubique cyclique'', Ph.D. Thesis, Univ. Laval, QC, 1995]. \par The last part of the paper comes from the doctoral dissertation of M. C. Ismaili [``Sur la capitulation des 3-classes d'idéaux de la clôture normale d'un corps cubique pur'', Ph.D. Thesis, Univ. Laval, QC, 1992] and concerns the principalization of 3-subgroups of ideal classes in unramified cyclic 3-extensions over the dihedral closure $K = \Bbb Q [\root3\of n,j]$ of a pure cubic field $\Bbb Q [\root3\of n]$ in the case where the 3-group of ideal classes ${\rm Cl}_K$ is isomorphic to $C_3 \times C_3$. \par In the three cases the action of the Galois group on the $p$-Sylow of the ideal class group [resp.\ for $p$ = 2 or 3] and the genus theory give rise to a natural typology of such fields including H. Kisilevsky's alternative [J. Number Theory {\bf 8} (1976), no.~3, 271--279; MR0417128 (54 \#5188)]: there are 3 types in the first and the last case and 2 types in the second case. Here the authors give a complete characterization of each type and they illustrate all cases by numerical examples.