Geodesic
Capitulation in Class Field Extensions of Type (p, p)
Canadian journal of mathematics, Tome 32 (1980) no. 5, pp. 1229-1243

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

Let K be a number field, K (1) its Hilbert class field, that is, the maximal abelian unramified extension of K, let K (2) be the Hilbert class field of K (1), and let G = Gal(K(2)/K) (alternatively, for p a prime the first and second p class fields enjoy properties analogous to those of the respective class fields discussed in this introduction; the particulars may be found surrounding Lemma 2). Since G/G’ is the largest abelian quotient of G, G/G′ = Gal (K(1))/K) and so G’ is the abelian group Gal(K (2)K(1)); moreover, class field theory provides (Artin) maps φK , φ K(1) which are isomorphisms of the class groups Ck, Ck(1) onto G/G′, G′ respectively. In the remarkable paper [1] E. Artin computed the composition V G′ where e is the homomorphism induced on the class groups by extending ideals of K to ideals of K (1), and he gave a formula for computing V G′ , the now familiar transfer (Verlagerung) homomorphism, in terms of the group G alone (see Lemma 1). 
Chang, S. M.; Foote, R. Capitulation in Class Field Extensions of Type (p, p). Canadian journal of mathematics, Tome 32 (1980) no. 5, pp. 1229-1243. doi: 10.4153/CJM-1980-091-9
@article{10_4153_CJM_1980_091_9,
     author = {Chang, S. M. and Foote, R.},
     title = {Capitulation in {Class} {Field} {Extensions} of {Type} (p, p)},
     journal = {Canadian journal of mathematics},
     pages = {1229--1243},
     year = {1980},
     volume = {32},
     number = {5},
     doi = {10.4153/CJM-1980-091-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-091-9/}
}
TY  - JOUR
AU  - Chang, S. M.
AU  - Foote, R.
TI  - Capitulation in Class Field Extensions of Type (p, p)
JO  - Canadian journal of mathematics
PY  - 1980
SP  - 1229
EP  - 1243
VL  - 32
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-091-9/
DO  - 10.4153/CJM-1980-091-9
ID  - 10_4153_CJM_1980_091_9
ER  - 
%0 Journal Article
%A Chang, S. M.
%A Foote, R.
%T Capitulation in Class Field Extensions of Type (p, p)
%J Canadian journal of mathematics
%D 1980
%P 1229-1243
%V 32
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-091-9/
%R 10.4153/CJM-1980-091-9
%F 10_4153_CJM_1980_091_9

[1] 1. Artin, E., Ideal Klassen in Oberkörpern und allgemeines Reziprozitatsgesetz, Abh. Math. Sem. Univ. Hamburg 7 (1930), 46–51. Google Scholar

[2] 2. Cassels, J. and Fröhlich, A., Algebraic number theory, Chapter IX (Academic Press, London, 1967). Google Scholar

[3] 3. Chang, S. M., Capitulation problems in algebraic number fields, Ph.D. thesis, University of Toronto (1977). Google Scholar

[4] 4. Furtwängler, P., Beweis der Hauptidealsatzes fur Klassenkörper algebraischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg 7 (1930), 14–36. Google Scholar

[5] 5. Gerth, F., Number fields with prescribed l-class groups, Proc. Amer. Math. Soc. 49 (1975), 284–288. Google Scholar

[6] 6. Gorenstein, D., Finite groups (Harper and Row, New York, 1968). Google Scholar

[7] 7. Hall, M., The theory of groups (Macmillan, New York, 1959). Google Scholar

[8] 8. Kisilevsky, H., Number fields with class number congruent to 4 mod 8 and Hilbert's Theorem 94, J. Number Theory 3 (1976), 271–279. Google Scholar

[9] 9. Matsumara, N., On the class field tower of an imaginary quadratic number field, Mem. Fac. Sci. Kyushu Univ. 31 (1977), 165–171. Google Scholar

[10] 10. Scholz, A. and Taussky, O., Die Hauptideale der Kubischen Klassenkörper imaginarquadratischer Zahlkörper, J. Reine Angew, Math. 171 (1934), 19–41. Google Scholar

[11] 11. Taussky, O., A remark concerning Hubert's Theorem 94, J. Reine Angew. Math. 239/240 (1970), 435–438. Google Scholar

[12] 12. Zassenhaus, H., The theory of groups (Chelsea, New York, 1949). Google Scholar

Cité par Sources :