I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) over $\mathbb{Q}$, as well as a separate proof of the theorem over all global fields.

In order to do this, I first want to compare all the proofs given in the existing literature. So far, I have managed to compile the following list of books and papers, in which proofs of the theorem can be found:

• Z.I. Borevich and I.R. Shafarevich: Number Theory (1964)
• J.W.S. Cassels: Rational Quadratic Forms (1978)
• J.W.S. Cassels: Lectures on Elliptic Curves (1991) [in which the theorem is only proved in the case of three variables]
• H. Hasse: Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen (1923); Über die Äquivalenz quadraticher Formen im Körper der rationalen Zahlen (1923); Symmetrische Matrizen im Körper der rationalen Zahlen (1924); Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper (1924) [the proof here extending through four papers and applying to all algebraic number fields]
• Y. Kitaoka: Arithmetic of Quadratic Forms (1993)
• O.T. O'Meara: Introduction to Quadratic Forms (1963) [in which the theorem is proved for all global fields]
• J.-P. Serre: Cours d'arithmétique (1970)
• G. Shimura: Arithmetic of Quadratic Forms (2010)

The proofs given in the above resources are all quite remarkably dissimilar in certain cases, which is encouraging, in as far as it suggests that the "canonical" proof of the theorem has yet to be established.

I would hence like to ask the members of the MathOverflow community if they are aware of any other proofs of the theorem, and if they could direct me to where they can be found.

• I am looking especially for proofs of the theorem in the more general case over all global fields, as I have so far only managed to find one proof of this (that in O'Meara).
• Proofs of the "weak" Hasse–Minkowski theorem (i.e. that pertaining to the equivalence of quadratic forms as opposed to their representing $0$) in the case of fields where the "strong" Hasse–Minkowski theorem does not hold are also especially welcome.
• Proof of the theorem for a particular $n \geq 3$ are also very welcome.
• The proofs do not have to be ones found in published books: Proofs from e.g. lectures notes are also very welcome, provided they are not entirely based on a proof given in a published book.

I intend to amend the above list as I find more proofs of the theorem.