Witt equivalence of semilocal Dedekind domains in global fields
Tóm tắt
We examine a condition for two semilocal Dedekind rings, the fields of fractions of which are global fields, to be Witt equivalent. To solve the problem we generalize the notion of a Hilbert-symbol equivalence introduced in [11] and prove that a Witt equivalence is equivalent to a Hilbert-symbol equivalence. As a result we describe a Witt equivalence in terms of field invariants.
Tài liệu tham khảo
R. Baeza,Quadratic Forms Over Semilocal Rings. LNM 655, Springer Verlag, Berlin-Heidelberg-New York, 1978.
J. P. Carpenter, Finiteness Theorems for Forms over Global Fields.Math. Z. 209 (1992), 153–166.
A. Czogała andA. Sładek, Higher Degree Hilbert Symbol Equivalence of Algebraic Number Fields, II.Journal of Number Theory 72 (1998), 363–376.
E. Hecke,Vorlesungen über die Theorie der Algebraischen Zahlen. Leipzig, 1923.
I. Kaplansky,Linear Algebra and Geometry. New York, Chelsea, 1974.
M. Knebusch, Grothendieck- und Wittringe von Nichtausgearteten Symmetrischen Bilinearformen. In:Sitzungsber. Heidelb. Akad. Wiss. Mat.-naturw. Kl., Vol.3, Springer Verlag, Berlin, 1969/1970.
T. Y. Lam,The Algebraic Theory of Quadratic Forms. Benjamin, Reading, Massachusetts, 1973.
D. B. Leep andA. R. Wadsworth, The Hasse Norm Theorem Mod Squares.J. Number Theory 42 (1992), 337–348.
M. A. Marshall,Bilinear Forms and Orderings on Commutative Rings. Queen’s Papers in Pure and Applied Mathematics — No. 71, Queen’s University Kingston, Ontario, Canada, 1985.
J. Milnor andD. Husemoller,Symmetric Bilinear Forms. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1973.
R. Perlis, K. Szymiczek, P. Conner, andR. Litherland, Matching Witts with Global Fields.Contemp. Math. 155 (1994), 365–387.
K. Szymiczek,Bilinear Algebra: an Introduction to the Algebraic Theory of Quadratic Forms. Algebra, Logic and Applications Series Vol. 7, Gordon and Breach Science Publishers, Amsterdam, 1997.
—, 2-ranks of Class Groups of Witt Equivalent Number Fields.Annales Mathematicae Silesianae 12 (1998), 53–64.
—, Ten Problems on Quadratic Forms.Acta Mathematica et Informatica Universitatis Ostraviensis 10 (2002), 133–143.
E. Witt, Teorie der Quadratischen Formen in Beliebigen Körpern.J. Reine Angew. Math. 176 (1937), 31–44.