Geodesic
Capitulation in unramified quadratic extensions of real quadratic number fields
Glasgow mathematical journal, Tome 36 (1994) no. 3, pp. 385-392

Voir la notice de l'article provenant de la source Cambridge University Press

Let k be an algebraic number field and Ck its ideal class group (in the wider sense). Suppose K is a finite extension of k. Then we say that an ideal class of k capitulates in K if this class is in the kernel of the homomorphisminduced by extension of ideals from k to K (See Section 2 below). In [4], Iwasawa gives examples of real quadratic number fields, with distinct primes Pi ≡ 1 (mod 4), for which all the ideal classes of the 2-class group, Ck,2 (the 2-Sylow subgroup of Ck), capitulate in an unramified quadratic extension of k. In these examples, Ck,2 is abelian of type (2,2), i.e. isomorphic to Z/2Z×Z/2Z and so all four ideal classes capitulate. 
Benjamin, E.; Sanborn, F.; Snyder, C. Capitulation in unramified quadratic extensions of real quadratic number fields. Glasgow mathematical journal, Tome 36 (1994) no. 3, pp. 385-392. doi: 10.1017/S0017089500031001
@article{10_1017_S0017089500031001,
     author = {Benjamin, E. and Sanborn, F. and Snyder, C.},
     title = {Capitulation in unramified quadratic extensions of real quadratic number fields},
     journal = {Glasgow mathematical journal},
     pages = {385--392},
     year = {1994},
     volume = {36},
     number = {3},
     doi = {10.1017/S0017089500031001},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500031001/}
}
TY  - JOUR
AU  - Benjamin, E.
AU  - Sanborn, F.
AU  - Snyder, C.
TI  - Capitulation in unramified quadratic extensions of real quadratic number fields
JO  - Glasgow mathematical journal
PY  - 1994
SP  - 385
EP  - 392
VL  - 36
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500031001/
DO  - 10.1017/S0017089500031001
ID  - 10_1017_S0017089500031001
ER  - 
%0 Journal Article
%A Benjamin, E.
%A Sanborn, F.
%A Snyder, C.
%T Capitulation in unramified quadratic extensions of real quadratic number fields
%J Glasgow mathematical journal
%D 1994
%P 385-392
%V 36
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500031001/
%R 10.1017/S0017089500031001
%F 10_1017_S0017089500031001

[1] 1.Cremona, J. E., and Odoni, R. W. K.. Some density results for negative Pell equations; an application of graph theory, J. London Math. Soc. (2) 39 (1989), 16–28. Google Scholar | DOI

[2] 2.Cremona, J. E. and Odoni, R. W. K.. A generalization of a result of Iwasawa on the capitulation problem, Math. Proc. Cambridge Phil. Soc. 107 (1990), 1–3. Google Scholar | DOI

[3] 3.Furtwängler, Ph., Über die Klassenzahl abelscher Zahlkörper, J. reine angew. Math. 134 (1908), 91–94. Google Scholar | DOI

[4] 4.Iwasawa, K., A note on the capitulation problem for number fields, Proc. Japan Acad. Ser. A. Math. Sci. 65 (1989), 59–61. Google Scholar

[5] 5.Janusz, G., Algebraic Number Fields, (Academic Press, New York, London, 1973). Google Scholar

[6] 6.Kubota, T., Über den bizyklischen biquadratischen Zahlkörpern, Nagoya Math. J. 10 (1956), 65–85. Google Scholar | DOI

[7] 7.Miyake, K., Algebraic investigations of Hilbert's Theorem 94, the principal ideal theorem, and the capitulation problem, Expo. Math. 7 (1989), 289–346. Google Scholar

[8] 8.Odoni, R. W. K., A note on a recent paper of Iwasawa on the capitulation problem, Proc. Japan Acad. Ser. A. Math. Sci. 65 (1989), 180–182. Google Scholar | DOI

[9] 9.Rosen, M., Two theorems on Galois cohomology, Proc. Amer. Math. Soc. 17 (1966), 1183–1185. Google Scholar | DOI

Cité par Sources :