The Witt Group of Degree k Maps and Asymmetric Inner Product Spaces
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914 Max L. Warshauer
The Witt Group of Degree k Maps and Asymmetric Inner Product Spaces
Springer-Verlag Berlin Heidelberg New York 1982
Author
Max L.Warshauer Department of Mathematics, Southwest Texas State University San Marcos, TX 78666, USA
AMS Subject Classifications (1980): 10 C 05
ISBN 3-540-11201-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11201-4 Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TABLE OF CONTENTS
INTRODUCTION .••••••.•....•••.•.•••.•..••..••......••........•
1
CONVENTIONS .•.•.•..•...•......•...•.........••......•..•..•••..
11
Chapter I
THE WITT RING .•.••••......•......•........•.......•
12
l.
Setting and notation ....••.•........•••........•..
12
2.
Inner products •.•.••••..•......•••...........••...
14
3.
Constructing new inner products out of old ....•...
23
4.
The symmetry operator .....••.........•..............
26
5.
The Witt equivalence relation ....•.•.•.•..•....••..
33
6.
Anisotropic representatives .••....•.••.••..•...•..
41
Chapter II
WITT INVARIANTS ..........•..•..•..•.•...•......
47
1.
Prime i d e a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.
Hilbert symbols •....•.......•..•..•................
51
3.
Rank. . . • . . . . . . . . • . • • . • . . . • . . • . . • . . . . . . . • . • . . . . . • . .
54
4.
Diagonalization and the discriminant .•........•..•
55
5.
Signatures.........................................
66
Chapter III Chapter IV
POLYNOMIALS........................ • . . . .. . . • . .
70
WITT GROUP OF A FIELD ...•.......•.••..•.••.....
79
l.
Decomposition by characteristic polynomial
79
2.
The trace lemma .•..........•••.............•....•.
90
3.
Computing Witt groups ......•.....•...............•. 100
4.
Torsion in
Chapter V
W(-k,Fl
.......•..•..•............•.....• 105
THE SQUARING MJ..P ..•..•..•..•.....•.......•...•..... 108
l.
Scharlau's transfer .•......•.••.....•..............•... III
2.
The exact octagon over a field . . . . . . . . . . . . . • . . . . . . 123
IV
Chapter VI
THE BOUNDARy •... , ...•.••.............•........•..... 133
1.
The boundary homomorphism
133
2.
Reducing to the maximal order
142
3.
Computing the local boundary
a(D,p)
for the
maximal order 4.
152
Computing the cokernel of
Chapter VII
a (D)
• . . . . . . . • . . . . . . . . . 167 184
NON MAXIMAL ORDERS
1.
Traces and canonical localizers .......•..•...•... 184
2.
Normal extensions
3.
Data Loading...
