Followup to question on $5$-adics, if $k \in \mathbb{Q}_5^\times$, is there $x_1, x_2, x_3 \in \mathbb{Q}_5$ where $\sum_{i = 1}^3 x_i^2 = k$? This is a followup to my question here.
My question is as follows. If $k \in \mathbb{Q}_5^\times$, then are there $x_1$, $x_2$, $x_3 \in \mathbb{Q}_5$ where$$x_1^2 + x_2^2 + 3x_3^2 = k?$$My idea is by using the fact that$$x_1^2 + x_2^2 + 3x_3^2 = k \implies (cx_1)^2 + (cx_2)^2 + 3(cx_3)^2 = c^2k$$we can possibly reduce the problem to the cases $k = 1$, $2$, $5$, $10$?
 A: You're correct: for any $c$ and $k$ in $\mathbb{Q}_5$, we can say that $k$ is a sum of three squares if and only if $c^2 k$ is a sum of three squares. Since $\{1,2,5,10\}$ is a set of coset representatives for $\mathbb{Q}_5^\times/(\mathbb{Q}_5^\times)^2$, every element of $\mathbb{Q}_5^\times$ equals one of those four numbers up to a multiple by a square, and thus we need only check those cases. (And those cases can be checked over $\mathbb{Z}$ mentally.)
A: Let $K$ be a finite extension of $\mathbf Q_p$ . Supposing $p$ odd, we have seen that $K^*/K^{*2}$ is a vector space of dimension 2 over $\mathbf F_2$ . For K = $\mathbf Q_5$ , you ask if the quadratic form $f = z^2 + x^2 + 3 y^2$ (I prefer to write it in this way) represents every element of K. Of course, knowing explicitly all the elements of $K$,  one just has to check numerically. But I guess that a theoretical (and generalizable) solution is preferable. I’ll refer for this to the masterful account given over $\mathbf Q_p$  in the first chapters of Serre’s book « A course in Arithmetic ». It is a general property that a quadratic form represents every element of K iff it represents 0. In chapter 3, Serre introduces the Hilbert symbol (a,b), which is a non degenerate bilinear form on $K^*/K^{*2} \times K^*/K^{*2}$ with values $\pm 1$, and (a,b) = 1 iff the quadratic form $z^2 -a x^2 -b y^2$ represents 0. In your example, a = -1, b = -3, which are units in $\mathbf Z_5$, and the formulas for Hilbert symbols (chapter 3, thm.1) immediately give what you want. All this can be generalized to general non singular quadratic forms over $\mathbf Q_p$, see chapter 4.
