Lam Introduction to Quadratic Forms over Fields declines to prove the Hasse-Minkowski Theorem in full (for forms over all number fields), saying "a full proof would usually involve some class field theory or else some deep arithmetic fact such as Dirichlet's theorem on primes in an arithmetic progression."
To judge from O'Meara 1973 Introduction to Quadratic Forms, though, Lam should have just said class field theory. O'Meara notes on p. 187n that Dirichlet's theorem on primes in progressions has been used for proofs over the rationals, but not arbitrary number fields.
O'Meara puts it this way:
Needless to say, it would be of great interest and importance to have
a direct proof of the entire theory (of quadratic forms on number
fields)
Has anyone later than that managed to prove the whole Hasse-Minkowski theorem using Dirichlet's theorem?
Best Answer
So far the answer to the question is no. Lam was speaking in very broad terms, and O'Meara's wish for a direct proof is unmet.
When Lam spoke of using the Dirichlet theorem on primes in progressions he may have meant proofs like Clark's Quadratic Forms over Global Fields
There the Global Square Theorem (saying a non-zero element of a global field is a square if it is a square at every place of the field) is quickly derived from the Cebotarev Density Theorem which is a generalization of Dirichlet's theorem.
The rest of that proof of Hasse-Minkowski uses the Hasse Norm Theorem which uses class field theory but this reasoning may be what Lam had in mind.