Number Theory: An Introduction to Mathematics

292 VII The Arithmetic of Quadratic Forms

characteristic. For any fieldF, we will denote byF×the multiplicative group of all
nonzero elements ofF. The squares inF×form a subgroupF×^2 and any coset of this
subgroup is called asquare class.
LetVbe a finite-dimensional vector space over such a fieldF. We say thatVis a
quadratic spaceif with each ordered pairu,vof elements ofVthere is associated an
element (u,v)ofFsuch that

(i)(u 1 +u 2 ,v)=(u 1 ,v)+(u 2 ,v)for allu 1 ,u 2 ,v∈V;
(ii)(αu,v)=α(u,v)for everyα∈Fand allu,v∈V;
(iii)(u,v)=(v,u)for allu,v∈V.

It follows that

(i)′(u,v 1 +v 2 )=(u,v 1 )+(u,v 2 )for allu,v 1 ,v 2 ∈V;
(ii)′(u,αv)=α(u,v)for everyα∈Fand allu,v∈V.

Lete 1 ,...,enbe a basis for the vector spaceV.Thenanyu,v∈Vcan be uniquely
expressed in the form

u=

∑n

j= 1

ξjej,v=

∑n

j= 1

ηjej,

whereξj,ηj∈F(j= 1 ,...,n),and

(u,v)=

∑n

j,k= 1

αjkξjηk,

whereαjk=(ej,ek)=αkj. Thus

(u,u)=

∑n

j,k= 1

αjkξjξk

is aquadratic formwith coefficients inF. The quadratic space is completely deter-
mined by the quadratic form, since

(u,v)={(u+v,u+v)−(u,u)−(v,v)}/ 2. (1)

Conversely, for a given basise 1 ,...,enof V,anyn×nsymmetric matrix
A=(αjk)with elements fromF, or the associated quadratic form f(x)=xtAx,
maybeusedinthiswaytogiveVthe structure of a quadratic space.
Lete′ 1 ,...,e′nbe any other basis forV.Then

ei=

∑n

j= 1

τjie′j,

whereT=(τij)is an invertiblen×nmatrix with elements fromF. Conversely, any
such matrixTdefines in this way a new basise′ 1 ,...,e′n.Since