Hasse-Minkowski

Local Global principle for quadratic forms over a field. Why is it true? How do the local solutions “force” Global solutions?

Let stick to rationals. What happens for variables? This case basically corresponds to understanding the form Having a non-trivial rational solution corresponds to being a square. And this can be checked locally! (Every non-square is not a square modulo some prime p-why?- even this fact is slightly tricky for other number fields)

What about ? Focus on Here we need ideas about Hilbert symbol- (bilinear properties of Hilbert symbol corresponds to multiplicativity of norms on quadratic extensions- basically when you multiply expression of the form X^2-DY^2 we get another expression of the same form!) – this fact can be used to reduce from to another form with smaller coefficients Keep doing this reduction and finally verify the local to Global principle for the few forms that you get at the end.

We can find $t_p$ non-zero mod p such that because we have solutions mod p for In general $t_p^k$ for all prime power $p^k.$ Construct a rational $t$ that locally looks like these $t_p^k.$- why does such a number exist? – (Apply Chinese Remainder theorem etc.) So we have locally. That is we have local solutions to By using the above argument for $n=3$ we get rational solutions to Scaling and subtracting t from these equations gives a non-trivial rational solution to our initial quadratic form. (some more work..)

-again split into two forms and reduce to smaller cases…

What is the source of getting global solutions from local solutions? 1. Being a square can be checked locally. 2. For n=3, we apply the reductions to solve equations with bigger coefficients by using equations with smaller coefficients. 3. What happens for n>4?