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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 algebraic number 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)
  • T.Y. Lam: The Algebraic Theory of Quadratic Forms (1973)
  • 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 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 books 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: The answers to this post have helped identify the following resources, in which the theorem is either proved in its entirety or otherwise meaningfully discussed in some manner (works labelled with an asterisk are ones of which I have thus far been unable to obtain a copy):

  • J.W.S. Cassels: Note on Quadratic Forms Over the Rational Field (1959)
  • L.E. Dickson: Studies in the Theory of Numbers (1930)*
  • M. Eichler: Quadratische Formen und orthogonale Gruppen (1952)*
  • A. Gamzon: The Hasse–Minkowski Theorem (2006) [an honours thesis written at the University of Connecticut under the supervision of Keith Conrad]
  • R. Heath-Brown: A New Form of the Circle Method, and its Application to Quadratic Forms (1996)
  • B.W. Jones: The Arithmetical Theory of Quadratic Forms (1950)
  • C.L. Siegel: Equivalence of Quadratic Forms (1948)
  • Th. Skolem: Diophantische Gleichungen (1938)*
  • T.A. Springer: Note on Quadratic Forms over Algebraic Number Fields (1957)*
  • G.L. Watson: Integral Quadratic Forms (1960)*
  • E. Witt: Theorie der quadratischen Formen in beliebigen Körpern (1937)

I have decided to keep the works identified after the creation of this post on this separate list. I am currently in the process of reviewing all of the above resources, and will continually be amending the list with new entries and commentary on existing entries.

Edit 30 July 2021: I have identified the following classical proofs of the theorem over $\mathbb{Q}$ in the case of $n=3$ (where the theorem is given in Legendre’s classical pre-$p$-adic formulation in terms of congruence conditions for the coefficients):

  • P.G.L. Dirichlet: Vorlesungen Über Zahlentheorie, 2. Aufl. (1871) [§§156-157]
  • C.F. Gauss: Disquisitiones Arithmeticae (1801) [§§293-298]
  • A.-M. Legendre: Essai sur la Théorie des Nombres (1798) [§IV]

For the proof of the theorem over all algebraic number fields, the $n=2$ case of course follows from the global square theorem over algebraic number fields, which was first proved in:

  • D. Hilbert: Über die Theorie des relativquadratischen Körpers (1898)

The $n=3$ case was originally proved by Furtwängler (albeit in a different formulation) in:

  • Ph. Furtwängler: Über die Reziprozitätsgesetze für ungerade Primzahlexponenten, Parts I-III (1909, 1912 and 1913 resp.)

These works were cited by Hasse in his:

  • H. Hasse: Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper (1924) [also listed above]

For $n=4$, there is of course Hasse’s original proof, the books of Lam and O’Meara and the following paper by Springer:

  • T.A. Springer: Note on Quadratic Forms over Algebraic Number Fields (1957) [also listed above]

Edit 10 August 2021: I will 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 know how to prove 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 is obviously equivalent to the quadratic case of the of the Hasse norm theorem, which was, as already mentioned, originally proved by Furtwängler in part III of the above listed series of papers. I 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.

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. - If there exists an online list of errata to Cassels' Rational Quadratic Forms, then 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)?

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    $\begingroup$ Adam Gamzon's senior thesis on HM in 2006 covers $\mathbf Q$ and $\mathbf F(T)$ for finite $\mathbf F$ of odd characteristic. Its Theorem 4.7 is a detour through number fields, showing (by a proof of Springer) that HM over number fields for $n = 3$ implies HM over number fields for $n = 4$. The proof for $n = 4$ over a number field involves $n = 3$ over a quadratic extension, so it's important in this proof to formulate it over number fields. The appendix has a cohomological proof that $[K^\times:{\rm N}_{L/K}(L^\times)] = 2$ for local fields not of char. $2$ (e.g., letting $K$ be $2$-adic). $\endgroup$
    – KConrad
    Feb 19, 2021 at 0:15
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    $\begingroup$ Here is a link to Gamzon's senior thesis: opencommons.uconn.edu/srhonors_theses/17. $\endgroup$
    – KConrad
    Feb 19, 2021 at 0:16
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    $\begingroup$ But this should hopefully give us some understanding of why the Dirichlet density theorem seems to be so indispensable to the proof for $n=4$ by offering some insight into its relation to the underlying analytic machinery (i.e. in terms of $L$-series)! $\endgroup$
    – user174434
    Feb 19, 2021 at 20:03
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    $\begingroup$ Hi good evening @JoséHdz.Stgo.! Well it depends: I would argue that the most insightful proof is that based on Minkowski's geometry of numbers, as given e.g. in Cassels' book Lectures on Elliptic Curves, but it relies on some familiarity with the theory of numbers. If one has no knowledge of the geometry of numbers, I would recommend going for the Borevich and Shafarevich proof first, as it provides the most detail. - The arguably briefest and cleverest proof for $n=3$ is that by Serre. $\endgroup$
    – user174434
    Feb 19, 2021 at 21:53
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    $\begingroup$ There is also something to be said for those of Kitaoka and Shimura (independent), which are both less elementary proofs that rely more heavily on the structural properties of quadratic forms/modules, but are absolutely excellent if that's what you're in to! $\endgroup$
    – user174434
    Feb 19, 2021 at 21:55

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Cassels also has a

Note on quadratic forms over the rational field. Proc. Cambridge Philos. Soc. 55 (1959), 267–270, addendum 57 (1961), 697.

https://doi.org/10.1017/S0305004100034009

There is perhaps some methodological interest in developing the theory of quadratic forms over the rational field using only the methods of elementary arithmetic. Hitherto it has appeared necessary to use theorems of a fairly deep nature, most often Dirichlet's theorem about the existence of primes in arithmetic progressions (e.g. Minkowski(1), Hasse(2), Dickson(8), Skolem(9), Burton Jones(6)). Skolem(5) uses a weaker form of Dirichlet's theorem which is rather easier to prove and Siegel(4) uses instead the machinery of the Hardy-Littlewood circle method. In this note I indicate how it is possible to develop the theory of quadratic forms over the rationals without using extraneous resources. Pall(10) states that he has also found such a development of the theory but he does not appear to have published it.

In the addendum, he regrets overlooking Eichler's version.

https://doi.org/10.1017/S0305004100035805

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    $\begingroup$ Oh this gives me an entire host of new resources! I will investigate them and report back with my findings! - I did find Cassels' references to other literature strangely wanting in his Rational Quadratic Forms, but here we have it! $\endgroup$
    – user174434
    Feb 18, 2021 at 22:37

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