A remark on arithmetic equivalence and the normset

Jim Coykendall

Acta Arithmetica (2000)

  • Volume: 92, Issue: 2, page 105-108
  • ISSN: 0065-1036

Abstract

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1. Introduction. Number fields with the same zeta function are said to be arithmetically equivalent. Arithmetically equivalent fields share much of the same properties; for example, they have the same degrees, discriminants, number of both real and complex valuations, and prime decomposition laws (over ℚ). They also have isomorphic unit groups and determine the same normal closure over ℚ [6]. Strangely enough, it has been shown (for example [4], or more recently [6] and [7]) that this does not imply that arithmetically equivalent fields are isomorphic. Just recently, B. De Smit and R. Perlis [3] showed that arithmetically equivalent fields do not even necessarily have the same class number. In this short note we take this recent result of De Smit and Perlis, and a well-known fact from algebraic number theory and use them to show that the integral normset (that is, the set of integral norms from a ring of algebraic integers to ℤ) uniquely determines a larger class of extensions of ℚ than the splitting set does. (In this paper, we use the standard convention that "equality of splitting sets" means that the symmetric difference is finite. That is to say, if X and Y are sets of prime ideals, then we say that they are equal if their symmetric difference is finite.) Good background information on the splitting set can be obtained in [1] or [5], and information on the normset can be found in [2].

How to cite

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Jim Coykendall. "A remark on arithmetic equivalence and the normset." Acta Arithmetica 92.2 (2000): 105-108. <http://eudml.org/doc/207372>.

@article{JimCoykendall2000,
abstract = { 1. Introduction. Number fields with the same zeta function are said to be arithmetically equivalent. Arithmetically equivalent fields share much of the same properties; for example, they have the same degrees, discriminants, number of both real and complex valuations, and prime decomposition laws (over ℚ). They also have isomorphic unit groups and determine the same normal closure over ℚ [6]. Strangely enough, it has been shown (for example [4], or more recently [6] and [7]) that this does not imply that arithmetically equivalent fields are isomorphic. Just recently, B. De Smit and R. Perlis [3] showed that arithmetically equivalent fields do not even necessarily have the same class number. In this short note we take this recent result of De Smit and Perlis, and a well-known fact from algebraic number theory and use them to show that the integral normset (that is, the set of integral norms from a ring of algebraic integers to ℤ) uniquely determines a larger class of extensions of ℚ than the splitting set does. (In this paper, we use the standard convention that "equality of splitting sets" means that the symmetric difference is finite. That is to say, if X and Y are sets of prime ideals, then we say that they are equal if their symmetric difference is finite.) Good background information on the splitting set can be obtained in [1] or [5], and information on the normset can be found in [2]. },
author = {Jim Coykendall},
journal = {Acta Arithmetica},
keywords = {arithmetically equivalent fields},
language = {eng},
number = {2},
pages = {105-108},
title = {A remark on arithmetic equivalence and the normset},
url = {http://eudml.org/doc/207372},
volume = {92},
year = {2000},
}

TY - JOUR
AU - Jim Coykendall
TI - A remark on arithmetic equivalence and the normset
JO - Acta Arithmetica
PY - 2000
VL - 92
IS - 2
SP - 105
EP - 108
AB - 1. Introduction. Number fields with the same zeta function are said to be arithmetically equivalent. Arithmetically equivalent fields share much of the same properties; for example, they have the same degrees, discriminants, number of both real and complex valuations, and prime decomposition laws (over ℚ). They also have isomorphic unit groups and determine the same normal closure over ℚ [6]. Strangely enough, it has been shown (for example [4], or more recently [6] and [7]) that this does not imply that arithmetically equivalent fields are isomorphic. Just recently, B. De Smit and R. Perlis [3] showed that arithmetically equivalent fields do not even necessarily have the same class number. In this short note we take this recent result of De Smit and Perlis, and a well-known fact from algebraic number theory and use them to show that the integral normset (that is, the set of integral norms from a ring of algebraic integers to ℤ) uniquely determines a larger class of extensions of ℚ than the splitting set does. (In this paper, we use the standard convention that "equality of splitting sets" means that the symmetric difference is finite. That is to say, if X and Y are sets of prime ideals, then we say that they are equal if their symmetric difference is finite.) Good background information on the splitting set can be obtained in [1] or [5], and information on the normset can be found in [2].
LA - eng
KW - arithmetically equivalent fields
UR - http://eudml.org/doc/207372
ER -

References

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  1. [1] J. W. S. Cassels and A. Frohlich, Algebraic Number Theory, Academic Press, London, 1967. Zbl0153.07403
  2. [2] J. Coykendall, Normsets and determination of unique factorization in rings of algebraic integers, Proc. Amer. Math. Soc. 124 (1996), 1727-1732. Zbl0856.11049
  3. [3] B. De Smit and R. Perlis, Zeta functions do not determine class numbers, Bull. Amer. Math. Soc. 31 (1994), 213-215. Zbl0814.11053
  4. [4] F. Gassmann, Bemerkungen zu der vorstehenden Arbeit von Hurwitz, Math. Z. 25 (1926), 124-143. 
  5. [5] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer and Polish Sci. Publ., Warszawa, 1990. Zbl0717.11045
  6. [6] R. Perlis, On the equation ζ_K(s) = ζ_K'(s), J. Number Theory 9 (1977), 342-360. Zbl0389.12006
  7. [7] R. Perlis, On the class numbers of arithmetically equivalent fields, ibid. 10 (1978), 489-509. Zbl0393.12009
  8. [8] R. Perlis and A. Schinzel, Zeta functions and the equivalence of integral forms, J. Reine Angew. Math. 309 (1979), 176-182. 

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