Offprint from PROCEEDINGS OF T H E L O N D O N M A T H E M A T I C A L SOCIETY Third Series
Volume XXXIV
January 1977
GENERIC SPLITTING OF QUADRATIC FORMS, II By MANFRED KNEBUSCH
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Third Series Editor-in-Chief G . R . A L L A N , School of Mathematics, The University, Leeds LS2 9JT Secretary j . E . R O S E B L A D B Executive Editor A L I C B C . S H A R P
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GENERIC SPLITTING OF QUADRATIC FORMS, II By M A N F R E D
KNEBUSCH
[Received 6 June 1975]
We present applications of the theory developed i n part I [5], studying forms with special properties in their splitting behaviour. The viewpoint of generic splitting is to a certain extent already implicitly contained i n the work of Arason, Pfister, Elman, and L a m (cf. the references i n [ 5 ] ) , and has led to important theorems. Nevertheless this viewpoint seems to generate many more problems than can be solved at this moment. I n my opinion an essential task of the present paper is to raise interest i n these problems, and I have written down some of them explicitly ( 4 . 1 3 , 6.7 i n part I, 8.3, 8.4, 10.6 i n part II). 7. Excellent forms Let be a non-split form over k, and let (K 0 < i < h) be a generic splitting tower of ) ( r ® % ) ) over hy with &(p)'&(r) the free composite of k( , whence a®T ~ 0. Choosing T = E(f) we see that a itself is certainly anisotropic. Now our study of a shows that a has height 1, and hence is isomorphic to the pure part of a Pfister form. But d i m a + 1 = dimr-f 2 is not a 2-power, since d i m r is a 2-power greater than 2. This is the desired contradiction, which proves that a ^ r®E. There always exists a place from E to F over k (Proposition 5.12). One may ask under which circumstances the fields E and F are actually equivalent. R E M A R K 9.5. Let (.£,, 0 ^ r ^ A) be a generic splitting tower of our odd-dimensional form
2 . Since Kr+\*L is the function field of ft® E^'L over E^'L and dim ft > d i m r for 0 < r < A - 2 , we see, again by Proposition 6.11, that all forms r®K *L with 0 < r < A - 1 are anisotropic. B u t T ® K _ * L splits. Thus r®L splits. r
h
x
r
h
x
In the special case where degp = 2 Theorem 9.6 yields the following corollary. C O R O L L A R Y 9.8. Let r be an anisotropic quaternion form over k, and let
