부산대학교 도서관

The authors independently study the capitulation (i.e. theprincipalization) of ideal classes in cyclic unramified extensions of degree 2 or 3 overcertain number fields of degree 2, 4 or 6.\par The first part of the paper is just a survey of A. Azizi's results onthe capitulationof 2-subgroups of ideal classes in unramified quadratic extensions overbiquadratic number fields $K = \Bbb Q [i, \sqrt d ]$ with 2-group ofideal classes isomorphic to $C_2 \times C_2$ [C. R. Acad. Sci. Paris Sér. IMath. {\bf 325} (1997), no.~2, 127--130; MR1467063 (98d:11131)].\par The second part of the paper under review is devoted to the study ofthe capitulation of the 3-subgroup of ideal classes in unramified cyclic3-extensions over absolute cubic cyclic number fields $K$ with 3-groupof ideal classes isomorphic to $C_3 \times C_3$. It comes from thedoctoral dissertation of M. Ayadi [``Sur la capitulation des 3-classesd'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 laclôture normale d'un corps cubique pur'',Ph.D. Thesis, Univ. Laval, QC, 1992]and concerns the principalization of 3-subgroups of ideal classes inunramified cyclic 3-extensions over the dihedral closure $K = \Bbb Q [\root3\ofn,j]$ of a pure cubic field $\Bbb Q [\root3\of n]$ in the case where the3-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 ofthe ideal class group [resp. for $p$ = 2 or 3] and the genus theorygive 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 secondcase. Here the authors give a complete characterization of each type and theyillustrate all cases by numerical examples.