Number Theory: An Introduction to Mathematics

1 Quadratic Spaces 297

Proof LetU+be a maximal positive definite subspace of the quadratic spaceV.Since
U+is certainly non-singular, we haveV=U+⊥W,whereW=U+⊥, and sinceU+is
maximal,(w,w)≤0forallw∈W.SinceU+⊆V,wehaveV⊥⊆W.IfU−is a
maximal negative definite subspace ofW, then in the same wayW=U−⊥U 0 ,where
U 0 =U−⊥∩W. EvidentlyU 0 is totally isotropic andU 0 ⊆V⊥. In factU 0 =V⊥,
sinceU−∩V⊥={ 0 }.Since(v,v)≥0forallv∈U+⊥V⊥, it follows thatU−is a
maximal negative definite subspace ofV.
IfU+′is another maximal positive definite subspace ofV,thenU+′∩W={ 0 }and
hence

dimU+′+dimW=dim(U+′ +W)≤dimV.

Thus dimU+′ ≤dimV−dimW=dimU+.ButU+andU+′can be interchanged. 

IfV is a quadratic space over an ordered fieldF,wedefinethepositive index

ind+Vto be the dimension of any maximal positive definite subspace. Similarly all

maximal negative definite subspaces havethe same dimension, which we will call the

negative indexofVand denote by ind−V. The proof of Proposition 9 shows that

ind+V+ind−V+dimV⊥=dimV.

Proposition 10Let F denote the real fieldRor, more generally, an ordered field in

which every positive element is a square. Then any non-singular quadratic form f in

n variables with coefficients from F is equivalent over F to a quadratic form

g=ξ 12 +···+ξ^2 p−ξ^2 p+ 1 −···−ξn^2 ,

where p∈{ 0 , 1 ,...,n}is uniquely determined by f. In fact,

ind+f=p,ind−f=n−p,indf=min(p,n−p).

Proof By Proposition 2,fis equivalent overFto a diagonal formδ 1 η^21 +···+δnηn^2 ,

whereδj= 0 ( 1 ≤j≤n). We may choose the notation so thatδj>0forj≤pand

δj<0forj>p. The change of variablesξj=δ^1 j/^2 ηj(j≤p),ξj =(−δj)^1 /^2 ηj

(j>p)now bringsfto the formg. Since the corresponding quadratic space has a

p-dimensional maximal positive definite subspace,p =ind+f is uniquely deter-

mined. Similarlyn−p=ind−f, and the formula for indffollows readily. 

It follows that, for quadratic spaces over a field of the type considered in Proposi-

tion 10, a subspace is anisotropic if and only if it is either positive definite or negative

definite.

Proposition 10 completely solves the problem of equivalence for real quadratic

forms. (The uniqueness ofpis known asSylvester’s law of inertia.) It will now be

shown that the problem of equivalence for quadratic forms over a finite field can also

be completely solved.

Lemma 11If V is a non-singular 2 -dimensional quadratic space over a finite field

Fq, of (odd) cardinality q , then V is universal.