According to "The Geometry of Four-Manifolds" by Donaldson and Kronheimer, indefinite unimodular forms are classified by their rank, signature and type. This is the Hasse-Minkowski classification of indefinite forms, they say.


However, this seems to be a bit of a folklore theorem, as I cannot find a single citation for it; all of my searches for any permutation of "hasse minkowski indefinite quadratic classification" yield instead a different theorem, namely one about solving the equation $Q(x) = r$ over $\mathbb{Q}$ for a given quadratic form $Q$.


Is it simply the case that the integral classification (which essentially states, for the case of an even form, that it is a sum of $E\_8$ lattices and Hyperbolics) is an easy consequence of this other theorem? I'm not familiar enough with quadratic forms to see how this should be so.


If it isn't an easy consequence, is there a reference for the integral classification that I just haven't found?



What is a reference for the Hasse-Minkowski classification of indefinite forms?
For a survey of the topic check out Milnor and Husemoller, Symmetric Bilinear Forms, Springer, 1973, II.3. They cite Borevich-Shafarevich (Number Theory), O'Meara (Introduction to Quadratic Forms), and Serre (Cours d'Arithmétique).


