R is injective. Proof: If i(a)=[a,OJ=i(b)=[b,OJ, then there is a c with a+O+c=b +O+c and hence a=h. 0 We want to apply the Grothendieck construction to W+(K). In order to obtain a ring W(K) we first define a product for symmetric bilinear spaces. To assume as little linear algebra as possible we use matrix language for this definition. Congruence of symmetric matrices (see chapter 1, § 2) will be denoted by ~.
If q:= 3 (mod 4) then -1 is not a square. Proof: IF; is a cyclic group of even order q -1. The only element of order 2, namely -1, is a square if and only if the order of IF; is a multiple of 4. That is exactly our claim. 10. Corollary. There is (up to isometry) exactly one anisotropic 2dimensional form over IFq. This is (1,e) ifq:=1(mod4) (1,1) if q:=3(mod4). 11. Corollary. There are group isomorphisms W(IFq)~71/271 x 7l/271 W(IFq) ~ 7l/471 if q:= 1 (mod 4) if q:= 3 (mod 4). The proof is left as an exercise to the reader.
An introduction to Diophantine approximation by J. W. S. Cassels