Index for subgroups of the group of units in number fields

• Autorzy: Tsutomu Shimada
• Języki publikacji: EN

Abstrakty

EN

We define a sequence of rational integers $u_i(E)$ for each finite index subgroup E of the group of units in some finite Galois number fields K in which prime p ramifies. For two subgroups E' ⊂ E of finite index in the group of units of K we prove the formula $v_p([E:E']) = ∑_{i=1}^r {u_i(E') - u_i(E)}$. This is a generalization of results of P. Dénes [3], [4] and F. Kurihara [5].

Kategorie tematyczne

• 11R21: Other number fields
• 11R29: Class numbers, class groups, discriminants

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1998
• Tom: 85
• Numer: 3
• Strony: 249-263

Daty

wydano

1998

otrzymano

1997-07-15

poprawiono

1998-03-03

Twórcy

autor

Tsutomu Shimada

• Kanagawa Prefectural Yamato-Higashi High-School, 1760 Fukami, Yamato, Kanagawa, 242, Japan

Bibliografia

• [1] A. Brumer, On the units of algebraic number fields, Mathematika 14 (1967), 121-124.
• [2] P. Dénes, Über irreguläre Kreiskörper, Publ. Math. Debrecen 3 (1953), 17-23.
• [3] P. Dénes, Über Grundeinheitssysteme der irregulären Kreiskörper von besonderen Kongruenzeigenschaften, Publ. Math. Debrecen 3 (1954), 195-204.
• [4] P. Dénes, Über den zweiten Faktor der Klassenzahl und den Irregularitätsgrad der irregulären Kreiskörper, Publ. Math. Debrecen 4 (1956), 163-170.
• [5] F. Kurihara, On the p-adic expansion of units of cyclotomic fields, J. Number Theory 32 (1989), 226-253.
• [6] L. C. Washington, Units of irregular cyclotomic fields, Illinois J. Math. 23 (1979), 635-647.
• [7] L. C. Washington, Introduction to Cyclotomic Fields, Springer, New York, 1997.