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THE DIMATU 
Hasse-Minkowski
theorem : let f(X,Y) Î Q[X,Y] be
a quadratic polynomial. Then the equation f(X,Y) = 0 has a solution (x,y) Î Q² if
and only if it has a solution (x,y)Î R² and a solution (x,y) Î Q²p
for every prime p. Here Qp is the field of p-adic number. You can find a demonstration of this theorem in the book of
Jean-Pierre Serre "A course in Arithmetic", Springer-Verlag.
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