3 The Hasse–Minkowski Theorem 317Proof Only the sufficiency of the condition requires proof. But if the rational quadratic
formf(ξ 1 ,...,ξn)representscinQvfor allvthen, by Theorem 36, the quadratic form
f∗(ξ 0 ,ξ 1 ,...,ξn)=−cξ 02 +f(ξ 1 ,...,ξn)is isotropic inQ. HencefrepresentscinQ, by Proposition 6.
Proposition 39Two non-singular quadratic forms with rational coefficients are equiv-
alent overQif and only if they are equivalent over all completionsQv.
Proof Again only the sufficiency of the condition requires proof. Letfandgbe non-
singular rational quadratic forms innvariables which are equivalent overQvfor allv.
Suppose first thatn=1andthatf =aξ^2 ,g=bη^2. By hypothesis, for everyv
there existstv∈Q×v such thatb=at^2 v. Thusba−^1 is a square inQ×v for everyv,and
henceba−^1 is a square inQ×, by part (i) of the proof of Theorem 36. Thereforefis
equivalent togoverQ.
Suppose now thatn>1 and the result holds for all smaller values ofn. Choose
somec∈Q×which is represented byfinQ.Thenfcertainly representscinQvand
hencegrepresentscinQv,sincegis equivalent tofoverQv. Since this holds for all
v, it follows from Corollary 38 thatgrepresentscinQ.
Thus, by the remark after the proof of Proposition 2,fis equivalent overQto a
quadratic formcξ 12 +f∗(ξ 2 ,...,ξn)andgis equivalent overQto a quadratic form
cξ 12 +g∗(ξ 2 ,...,ξn).Sincefis equivalent togoverQv, it follows from Witt’s can-
cellation theorem thatf∗(ξ 2 ,...,ξn)is equivalent tog∗(ξ 2 ,...,ξn)overQv.Since
this holds for everyv, it follows from the induction hypothesis thatf∗is equivalent to
g∗overQ,andsofis equivalent togoverQ.
Corollary 40Two non-singular quadratic forms f and g in n variables with rational
coefficients are equivalent over the rational fieldQif and only if
(i)(detf)/(detg)is a square inQ×,
(ii) ind+f=ind+g,
(iii)sp(f)=sp(g)for every prime p.
Proof This follows at once from Proposition 39, on account of Propositions 10
and 34.
Thestrong Hasse principle(Theorem 36) says that a quadratic form isisotropic
over the global fieldQif (and only if) it is isotropic over all its local completionsQv.
The so-namedweak Hasse principle(Proposition 39) says that two quadratic forms are
equivalentoverQif (and only if) they are equivalent over allQv.Theselocal-global
principleshave proved remarkably fruitful. They organize the subject, they can be
extended to other situations and, even when they fail, they are still a useful guide. We
describe some results which illustrate these remarks.
As mentioned at the beginning of this section, the strong Hasse principle continues
to hold when the rational field is replaced byany algebraic number field. Waterhouse
(1976) has established the weak Hasse principle for pairs of quadratic forms: if over
every completionQvthere is a change of variables taking bothf 1 tog 1 andf 2 tog 2 ,
then there is also such a change of variables overQ. For quadratic forms over the field