Witt equivalence of global fields. II. Relative quadratic extensions

Author:
Kazimierz Szymiczek

Journal:
Trans. Amer. Math. Soc. **343** (1994), 277-303

MSC:
Primary 11E12; Secondary 11E08, 11E81

DOI:
https://doi.org/10.1090/S0002-9947-1994-1176087-X

MathSciNet review:
1176087

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper explores the consequences of the Hasse Principle for Witt equivalence of global fields in the case of relative quadratic extensions. We are primarily interested in generating the Witt equivalence classes of quadratic extensions of a given number field, and we study the structure of the class, the number of classes, and the structure of the set of classes. Along the way, we reprove several results obtained earlier in the absolute case of the rational ground field, giving unified and short proofs based on the Hasse Principle.

**[C]**J. Carpenter,*Finiteness theorems for forms over global fields*, Math. Z.**209**(1992), 153-166. MR**1143220 (93g:11031)****[Cz1]**A. Czogala,*On recïprocity equivalence of quadratic number fields*, Acta Arith.**58**(1991), 27-46. MR**1111088 (92h:11090)****[Cz2]**-,*Witt equivalence of quadratic extensions of global fields*, Math. Slovaca**41**(1991), 251-255. MR**1126661 (92j:11034)****[E-L-W]**R. Elman, T. Y. Lam, and A. R. Wadsworth,*Quadratic forms under multiquadratic extensions*, Indag. Math. (N.S.)**42**(1980), 131-145. MR**577569 (81j:10028)****[J-M]**S. Jakubec and F. Marko,*Witt equivalence classes of quartic number fields*, Math. Comp.**58**(1992), 355-368. MR**1094952 (92e:11031)****[L]**T. Y. Lam,*The algebraic theory of quadratic forms*, Benjamin/Cummings, Reading, MA, 1980. MR**634798 (83d:10022)****[O'M]**O. T. O'Meara,*Introduction to quadratic forms*, Springer-Verlag, Berlin, Heidelberg, and New York, 1971. MR**0347768 (50:269)****[P-S-C-L]**R. Perlis, K. Szymiczek, P. E. Conner, and R. Litherland,*Matching Witts with global fields*, Recent Advances in Real Algebraic Geometry and Quadratic Forms (Proc. RAGSQUAD Year, Berkeley, CA, 1990-1991); (W. B. Jacob, T. Y. Lam, and R. O. Robson, eds.), Contemp. Math. (to appear). MR**1260697 (94h:14003)****[S1]**K. Szymiczek,*Matching Witts locally and globally*, Math. Slovaca**41**(1991), 315-330. MR**1126669 (92g:11039)****[S2]**-,*Witt equivalence of global fields*, Comm. Algebra**19**(1991), 1125-1149. MR**1102331 (92d:11031)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
11E12,
11E08,
11E81

Retrieve articles in all journals with MSC: 11E12, 11E08, 11E81

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1176087-X

Keywords:
Witt equivalence,
relative quadratic extensions,
Hasse principle

Article copyright:
© Copyright 1994
American Mathematical Society