# Proof of Hasse-Minkowski over Number Field

Can anyone point me towards a good reference which contains the proof of the Hasse-Minkowski theorem of quadratic forms over a number field? Serre's "A Course in Arithmetic" has a self contained proof for the theorem when the base field is $\mathbb{Q}$, but I am looking for the more general setting.

## 3 Answers

The proof of Hasse-Minkowski for more general global fields can be found in O'Meara, Introduction to Quadratic Forms, Springer-Verlag, 1973.

My memory is that this is given as an exercise in Cassels and Frolich (the exercises were actually written by Serre and Tate). One uses the fact that being a norm in a cyclic extension of number fields can be detected locally (the so-called Hasse norm theorem), which deals with the case of representations by binary quadratic forms. Forms in more variables are then handled by induction. (Possibly I'm simplifying here, but this is the basic idea.)

If I understand correctly, this is true to history: Hasse's contribution to the theorem was to prove it over number fields, using his norm theorem for cyclic extensions as the initial input.

• It is Exercise 4. "Numbers Represented by Quadratic Forms" in Cassels and Frohlich. – Colin McLarty Jan 9 '17 at 0:43

Other than O'Meara (pronounce O Marra, who knew?), Lam (2005), Introduction to Quadratic Forms over Fields, gives a proof for global fields modulo two assumed results on page 171.

Theorem 3.7 says that $a$ is a square in the global field if and only if it is a square at each place. I think that's only fair.

Theorem 3.8 says that a quaternion algebra splits if and only if it splits at each place. He mentions that this may be regarded as either a special case of either the Brauer-Hasse-Noether Theorem or as a special case of the Hasse Norm Theorem.

Cassels, Rational Quadratic Forms, points out on page 96 that there are fields where the Weak Hasse Principle holds but the Strong Hasse Principle does not. Then he sketches a proof for number fields in a page.

Cassels does use Dirichlet on primes in arithmetic progressions for Chapter 6, and Lam mentions that assuming that would be one way to deal with number fields; so I suggest that as a good exercise for a graduate student, see how far you get with Lam's treatment plus Dirichlet on primes.

Oh, try Shimura's 2010 book Arithmetic of Quadratic Forms if you want to do things like prove the Mass Formula.