On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields

• Autorzy: P. E. Conner , J. Hurrelbrink
• Języki publikacji: EN

Abstrakty

EN

A large number of papers have contributed to determining the structure of the tame kernel $K₂𝓞_F$ of algebraic number fields F. Recently, for quadratic number fields F whose discriminants have at most three odd prime divisors, 4-rank formulas for $K₂𝓞_F$ have been made very explicit by Qin Hourong in terms of the indefinite quadratic form x² - 2y² (see [7], [8]).
We have made a successful effort, for quadratic number fields F = ℚ (√(±p₁p₂)), to characterize in terms of positive definite binary quadratic forms, when the 2-Sylow subgroup of the tame kernel of F is elementary abelian.
This makes determining exactly when the 4-rank of $K₂𝓞_F$ is zero, computationally even more accessible. For arbitrary algebraic number fields F with 4-rank of $K₂𝓞_F$ equal to zero, it has been pointed out that the Leopoldt conjecture for the prime 2 is valid for F, compare [6].
We consider this paper to be an addendum to the Acta Arithmetica publications [7], [8]. It grew out of our circulated 1989 notes [3].
Acknowledgements. We gratefully acknowledge fruitful long-term communications on this topic with Jerzy Browkin.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1995
• Tom: 73
• Numer: 1
• Strony: 59-65

Daty

wydano

1995

otrzymano

1994-12-27

poprawiono

1995-02-18

Twórcy

autor

P. E. Conner

• Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803, U.S.A.

autor

J. Hurrelbrink

• Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803, U.S.A.

Bibliografia

• [1] P. Barrucand and H. Cohn, Note on primes of type x² + 32y², class number and residuacity, J. Reine Angew. Math. 238 (1969), 67-70.
• [2] P. E. Conner and J. Hurrelbrink, Class Number Parity, Ser. Pure Math. 8, World Sci., Singapore, 1988.
• [3] P. E. Conner and J. Hurrelbrink, Examples of quadratic number fields with K₂𝓞 containing no element of order four, circulated notes, 1989.
• [4] P. E. Conner and J. Hurrelbrink, The 4-rank of K₂𝓞, Canad. J. Math. 41 (1989), 932-960.
• [5] J. Hurrelbrink, Circulant graphs and 4-ranks of ideal class groups, J. Math. 46 (1994), 169-183.
• [6] M. Kolster, Remarks on étale K-theory and the Leopoldt conjecture, in: Séminaire de Théorie des Nombres, Paris, 1991-92, Progr. Math. 116, Birkhäuser, 1993, 37-62.
• [7] H. Qin, The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields, Acta Arith. 69 (1995), 153-169.
• [8] H. Qin, The 4-rank of $K₂O_F$ for real quadratic fields, Acta Arith. 72 (1995), 323-333.
• [9] B. A. Venkov, Elementary Number Theory, Wolters-Noordhoff, Groningen, 1970