Squares in p-adic integers What are the solutions of equation
$$x^2+y^2+z^2+w^2=0,$$
in the $p$-adic integers? I think that for $p = 2$ it has only the trivial solution, but for $p$ odd there are nontrivial solutions. 
 A: Well, there if $x_0^2\equiv a\pmod p$ then there is a solution to $x^2=a$ in $p$-adic solutions. So taking an integer solution $x_0^2+y_0^2+z_0^2+w_0^2=p$, with $x_0\neq 0$, we can find a $p$-adic solution to $x^2+y_0^2+z_0^2+w_0^2=p$.
We can certainly find infinitely many different answers by mutliplying $(x,y_0,z_0,w_0)$ by any $p$-adic number, but the real question is how many projective solutions.
A: This answer may be disproportionate to the context of the question, but, if by chance one knows about local behavior of quaternion algebras... one might recognize that quaternary quadratic form as being the norm-form of the (usual rational form of) the Hamiltonian quaternions. The only ramified places (at which that quaternion algebra remains a division algebra) are $2$ and $\mathbb R$. Thus, at all odd $p$ that form is isotropic, and at $2$ (and $\mathbb R$, obviously) it is anisotropic.
(Yes, certainly, Hensel's lemma is used in setting up the usual properties of  division algebras over local fields, as in Weil's "Basic Number Theory". Also, my on-line notes at http://www.math.umn.edu/~garrett/m/algebra/algebras.pdf do things similar to Weil, ...)
