Index for subgroups of the group of units in number fields

Tsutomu Shimada

Acta Arithmetica (1998)

  • Volume: 85, Issue: 3, page 249-263
  • ISSN: 0065-1036

Abstract

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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].

How to cite

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Tsutomu Shimada. "Index for subgroups of the group of units in number fields." Acta Arithmetica 85.3 (1998): 249-263. <http://eudml.org/doc/207167>.

@article{TsutomuShimada1998,
abstract = {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 $v_p([E:E^\{\prime \}]) = ∑_\{i=1\}^r \{u_i(E^\{\prime \}) - u_i(E)\}$. This is a generalization of results of P. Dénes [3], [4] and F. Kurihara [5].},
author = {Tsutomu Shimada},
journal = {Acta Arithmetica},
keywords = {cyclotomic units; -character of the Bernoulli numbers; index of subgroups; unit group of finite Galois number fields},
language = {eng},
number = {3},
pages = {249-263},
title = {Index for subgroups of the group of units in number fields},
url = {http://eudml.org/doc/207167},
volume = {85},
year = {1998},
}

TY - JOUR
AU - Tsutomu Shimada
TI - Index for subgroups of the group of units in number fields
JO - Acta Arithmetica
PY - 1998
VL - 85
IS - 3
SP - 249
EP - 263
AB - 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 $v_p([E:E^{\prime }]) = ∑_{i=1}^r {u_i(E^{\prime }) - u_i(E)}$. This is a generalization of results of P. Dénes [3], [4] and F. Kurihara [5].
LA - eng
KW - cyclotomic units; -character of the Bernoulli numbers; index of subgroups; unit group of finite Galois number fields
UR - http://eudml.org/doc/207167
ER -

References

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  1. [1] A. Brumer, On the units of algebraic number fields, Mathematika 14 (1967), 121-124. Zbl0171.01105
  2. [2] P. Dénes, Über irreguläre Kreiskörper, Publ. Math. Debrecen 3 (1953), 17-23. 
  3. [3] P. Dénes, Über Grundeinheitssysteme der irregulären Kreiskörper von besonderen Kongruenzeigenschaften, Publ. Math. Debrecen 3 (1954), 195-204. Zbl0058.26902
  4. [4] P. Dénes, Über den zweiten Faktor der Klassenzahl und den Irregularitätsgrad der irregulären Kreiskörper, Publ. Math. Debrecen 4 (1956), 163-170. Zbl0071.26505
  5. [5] F. Kurihara, On the p-adic expansion of units of cyclotomic fields, J. Number Theory 32 (1989), 226-253. Zbl0689.12005
  6. [6] L. C. Washington, Units of irregular cyclotomic fields, Illinois J. Math. 23 (1979), 635-647. Zbl0427.12004
  7. [7] L. C. Washington, Introduction to Cyclotomic Fields, Springer, New York, 1997. Zbl0966.11047

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