The cyclic subfield integer index

Bart de Smit

Journal de théorie des nombres de Bordeaux (2000)

  • Volume: 12, Issue: 1, page 209-218
  • ISSN: 1246-7405

Abstract

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In this note we consider the index in the ring of integers of an abelian extension of a number field K of the additive subgroup generated by integers which lie in subfields that are cyclic over K . This index is finite, it only depends on the Galois group and the degree of K , and we give an explicit combinatorial formula for it. When generalizing to more general Dedekind domains, a correction term can be needed if there is an inseparable extension of residue fields. We identify this correction term for abelian extensions of type ( p , p ) .

How to cite

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de Smit, Bart. "The cyclic subfield integer index." Journal de théorie des nombres de Bordeaux 12.1 (2000): 209-218. <http://eudml.org/doc/248486>.

@article{deSmit2000,
abstract = {In this note we consider the index in the ring of integers of an abelian extension of a number field $K$ of the additive subgroup generated by integers which lie in subfields that are cyclic over $K$. This index is finite, it only depends on the Galois group and the degree of $K$, and we give an explicit combinatorial formula for it. When generalizing to more general Dedekind domains, a correction term can be needed if there is an inseparable extension of residue fields. We identify this correction term for abelian extensions of type $(p, p)$.},
author = {de Smit, Bart},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Fitting ideal; Dedekind domain; finite abelian extension},
language = {eng},
number = {1},
pages = {209-218},
publisher = {Université Bordeaux I},
title = {The cyclic subfield integer index},
url = {http://eudml.org/doc/248486},
volume = {12},
year = {2000},
}

TY - JOUR
AU - de Smit, Bart
TI - The cyclic subfield integer index
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 209
EP - 218
AB - In this note we consider the index in the ring of integers of an abelian extension of a number field $K$ of the additive subgroup generated by integers which lie in subfields that are cyclic over $K$. This index is finite, it only depends on the Galois group and the degree of $K$, and we give an explicit combinatorial formula for it. When generalizing to more general Dedekind domains, a correction term can be needed if there is an inseparable extension of residue fields. We identify this correction term for abelian extensions of type $(p, p)$.
LA - eng
KW - Fitting ideal; Dedekind domain; finite abelian extension
UR - http://eudml.org/doc/248486
ER -

References

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  10. [10] H.W. Lenstra, Jr., Grothendieck groups of abelian group rings. J. Pure Appl. Algebra20 (1981), 173-193. Zbl0467.16016MR601683
  11. [11] C. Parry, Bicyclic bicubic fields. Canad. J. Math.42 (1990) no. 3, 491-507. Zbl0715.11059MR1062741
  12. [12] L. Rédei, Über das Kreisteilungspolynom. Acta Math. Hungar.5 (1954), 27-28. Zbl0055.01305MR62760
  13. [13] J.-P. Serre, Local fields. Springer-Verlag, New York, 1979. Zbl0423.12016MR554237
  14. [14] L.C. Washington, Introduction to cyclotomic fields. Springer-Verlag, New York, 1982. Zbl0484.12001MR718674

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