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 globalalgebraic number fields.
- I am looking especially for proofs of the theorem in the more general case over all algebraic number fields, as, apart from Hasse's original, the only full proofs of this that I have been able to find are those in the grand treatisesbooks of Lam (who proves it modulo two assumptions - the latter of which is very substantial) and O'Meara (who only proves the weak Hasse-Minkowski theorem).
- 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.
- Proofs of the theorem for a particular $n \geq 3$ are also very welcome ($n$ here denoting the number of variables of the quadratic forms).
- The proofs do not have to be ones found in published books: Proofs from e.g. lecture notes are also very welcome, provided they are not entirely based on a proof given in a published book.
Edit 10 August 2021: I have stillwill here give a sketch of the proof of the Hasse-Minkowski theorem over all global fields for all $n$ except $n=3$:
For $n=2$, this of course follows, as I mentioned above, from the global square theorem, which, in the case of algebraic number fields, follows from the Chebotarev density theorem. This appears as a corollary to theorem 10 of:
- S. Lang: Algebraic Number Theory, 2nd edition (1994)
and as exercise 6.2 of:
- Cassels and Fröhlich (editors): Algebraic Number Theory (1967)
I do not managedknow how to obtain a copyprove the global square theorem for function fields over finite fields (the other kind of global fields), but then this is not relevant to my project.
For $n=4$, Hasse originally proved it in a way very similar to the traditional proof of the theorem over $\mathbb{Q}$ for $n=4$, but the nicest way of proving the statement is by Springer's trick, first published in the paper listed above, and reproduced as exercise 4. –4 of Cassels and Fröhlich, which reduces the statement to the $n=3$ case.
For $n \geq 5$, Serre's topological argument for the theorem over $\mathbb{Q}$ from his Cours d'arithmétique generalises readily (almost verbatim) to all algebraic number fields.
I will now try to discuss the only problematic case, $n=3$: It appearsis obviously equivalent to have been publishedthe quadratic case of the of the Hasse norm theorem, which was, as already mentioned, originally proved by Furtwängler in a journal called Indagationes Mathematicaepart III of the above listed series of papers. If anyone happensI have the following to say:
First, a historical remark: In the Wikipedia article on the Hasse norm theorem, there is the following:
The full theorem is due to Hasse (1931). The special case when the degree $n$ of the extension is $2$ was proved by Hilbert (1897), and the special case when $n$ is prime was proved by Furtwangler (1902).
This is definitely wrong - the date 1897 seems to refer to Hilbert's Zahlbericht, where the theorem was not proved. As I have already noted, the quadratic case was first proved by Furtwängler. The special case when the degree of the extension is prime was proved by Hasse in:
- H. Hasse: Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Teil II (1930)
and the case of a general cyclic extension was later proved by Hasse in:
- H. Hasse (1931): Beweis eines Satzes und Wiederlegung einer Vermutung über das allgemeine Normenrestsymbol (1931)
I don't know how to edit Wikipedia articles, but if anyone here does, it would be good if we could correct this journal or.
As another aside: On p. 96, ch. 6 of Rational Quadratic Forms, when sketching the proof of the Hasse-Minkowski theorem over all global fields, Cassels makes the incorrect claim that the Hasse norm theorem holds for all abelian extensions - this is exactly the Vermutung that is widerlegt in Hasse's 1931 paper specifically has been digitalised somewhere. - If there exists an online list of errata to Cassels' Rational Quadratic Forms, all information would be greatly appreciatedthen this belongs there.
The above two remarks were not directly relevant to the proof of the Hasse-Minowski theorem, but the rest of what I have to say is: The Brauer-Hasse-Noether theorem is derived as a corollary of a high-level class field theoretic result in §9.6 of ch. VII of Cassels and Fröhlich, which in turn gives us the Hasse norm theorem and hence Hasse-Minkowski for $n=3$. Moreover, the Hasse norm theorem is also proved in:
- G. Janusz: Algebraic Number Fields (1973)
My question to the community is: Are there any other textbook proofs of the Hasse norm theorem, and are there any simpler proofs of the of theorem in the special case of quadratic extension (i.e. Furtwängler's result)?
