3 The Hasse–Minkowski Theorem 313was inspired. The assumption was first proved by Dirichlet (1837) and will be
referred to here as ‘Dirichlet’s theorem on primes in an arithmetic progression’. In
the present chapter Dirichlet’s theorem will simply be assumed, but it will be proved
(in a quantitative form) in Chapter X.
It was shown by Meyer (1884), although the published proof was incomplete, that
a quadratic form in five or more variables with integer coefficients is isotropic if it is
neither positive definite nor negative definite.
The preceding results are all special cases of theHasse–Minkowski theorem,which
is the subject of this section. LetQdenote the field of rational numbers. By Ostrowski’s
theorem (Proposition VI.4), the completionsQvofQwith respect to an arbitrary ab-
solute value||vare the fieldQ∞=Rof real numbers and the fieldsQpofp-adic
numbers, wherepis an arbitrary prime. The Hasse–Minkowski theorem has the
following statement:
A non-singular quadratic form f(ξ 1 ,...,ξn)with coefficients fromQis isotropic
inQif and only if it is isotropic in every completion ofQ.
This concise statement contains, and tosome extent conceals, a remarkable amount
of information. (Its equivalence to Legendre’s theorem whenn=3 may be established
by elementary arguments.) The theorem was first stated and proved by Hasse (1923).
Minkowski (1890) had derived necessary and sufficient conditions for the equivalence
overQof two non-singular quadratic forms with rational coefficients by using known
results on quadratic forms with integer coefficients. The role ofp-adic numbers was
taken by congruences modulo prime powers. Hasse drew attention to the simplifica-
tions obtained by studying from the outset quadratic forms over the fieldQ,rather
than the ringZ, and soon afterwards (1924) he showed that the theorem continues to
hold if the rational fieldQis replaced by an arbitrary algebraic number field (with its
corresponding completions).
The condition in the statement of the theorem is obviously necessary and it is only
its sufficiency which requires proof. Before embarking on this we establish one more
property of the Hilbert symbol for the fieldQof rational numbers.
Proposition 35For any a,b∈Q×, the number of completionsQvfor which one has
(a,b)v=− 1 (wherevdenotes either∞or an arbitrary prime p) is finite and even.
Proof By Proposition 23, it is sufficient to establish the result whenaandbare
square-free integers such thatabis also square-free. Then(a,b)r = 1forany
odd prime∏ rwhich does not divideab, by Proposition 25. We wish to show that
v(a,b)v =1. Since the Hilbert symbol is multiplicative, it is sufficient to estab-
lish this in the following special cases: fora=−1andb=− 1 , 2 ,p;fora=2and
b=p;fora=pandb=q,wherepandqare distinct odd primes. But it follows
from Propositions 24, 25 and 27 that
∏v(− 1 ,− 1 )v=(− 1 ,− 1 )∞(− 1 ,− 1 ) 2 =(− 1 )(− 1 )= 1 ;
∏v(− 1 , 2 )v=(− 1 , 2 )∞(− 1 , 2 ) 2 = 1 · 1 = 1 ;