Geodesic
Kummer's and Iwasawa's Version of Leopoldt's Conjecture
Canadian mathematical bulletin, Tome 31 (1988) no. 3, pp. 338-346

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

We present a refinement of Iwasawa's approach to Leopoldt's conjecture on the non-vanishing of the p-adic regulator of an algebraic number field K. As an application, the conjecture for K implies the conjecture for a solvable extension L of degree g over K if g is relatively prime to p — 1 and p does not divide g, the discriminant of K, and the quotient of class numbers where is a primitive pth root of unity. This can be viewed as generalizing a theorem of Kummer on cyclotomic units.
DOI : 10.4153/CMB-1988-049-0
Mots-clés : 11R27 
Sands, Jonathan W. Kummer's and Iwasawa's Version of Leopoldt's Conjecture. Canadian mathematical bulletin, Tome 31 (1988) no. 3, pp. 338-346. doi: 10.4153/CMB-1988-049-0
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[1] 1. Atiyah, M. F. and McDonald, I. G., Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass. 1969. Google Scholar

[2] 2. Ax, J., On the units of an algebraic number field, Illinois J. Math. 9 (1965), pp. 584–589. Google Scholar

[3] 3. Bertrandias, F. and Payan, J. J., T-extensions et invariants cyclotomiques, Ann. Scient. Ec. Norm. Sup. 4e ser. 5 (1972), pp. 517–543. Google Scholar

[4] 4. Brumer, A., On the units of algebraic number fields, Mathematika 14 (1967), pp. 121–124. Google Scholar

[5] 5. Chevalley, C., Deux théorèmes d'arithmétique, J. Math. Soc. Japan 31 (1951), pp. 36–44. Google Scholar

[6] 6. Gillard, R., Formulations de la conjecture de Leopoldt et étude d'une condition sufissante, Abh. Math. Sem. Univ. Hamburg 48 (1979), pp. 125–138. Google Scholar

[7] 7. Gras, G., Remarques sur la conjecture de Leopoldt, C.R. Acad. Se. Paris (A) 274 (1972), pp. 377–380. Google Scholar

[8] 8. Gras, G., Groupe de Galois de la p-extension abélienne p-ramifiée maximale d'un corps de nombres, J. Reine Angew. Math. 333 (1982), pp. 86–132. Google Scholar

[9] 9. Hecke, E., Lectures on the Theory of Algebraic Numbers, Springer-Verlag, New York, 1981. Google Scholar

[10] 10. Iwasawa, K., A simple remark on Leopoldt's conjecture, (in Japanese), R.I.M.S. Kyoto U. (1984), pp. 45–54. Google Scholar

[11] 11. Long, R., Algebraic Number Theory, Marcel Dekker, New York, 1977. Google Scholar

[12] 12. Miki, H. and Sato, H., Leopoldt's conjecture and Reiner's theorem, J. Math. Soc. Japan 361 (1984), pp. 47–51. Google Scholar

[13] 13. Miki, H., On the Leopoldt conjecture on the p-adic regulators, J. Number Theory 26 (1987), pp. 117–128. Google Scholar

[14] 14. Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, P.W.N. Polish Scientific Publishers, Warsaw, 1973. Google Scholar

[15] 15. Serre, J. P., Local Fields, Springer-Verlag, New York, 1979. Google Scholar

[16] 16. Washington, L., Class numbers and Z-extensions, Math. Ann. 214 (1975), pp. 177–193. Google Scholar

[17] 17. Washington, L., The non-p-part of the class number in a cyclotomic Z-extension, Inv. Math. 49 (1979), pp. 87–97. Google Scholar

[18] 18. Washington, L., Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982. Google Scholar

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