Determine whether the form has a non-trivial zero. (You do not need to exhibit it).
I know that I need to use the Hasse-Minkowski Theorem somehow. From examples I've seen online, I can split finding roots into a couple of cases.
I know that if I can find a root in each case, then there is a non-trivial zero in $\mathbb Q$. I also know that if there is a case in which I can't find a non-trivial zero, then there is no non-trivial zero in $\mathbb Q$.
Case 1: $\mathbb Q_p,p\nmid 2,3$ I used Hensel's Lemma and found a solution here.
Case 2: $\mathbb Q_2$ I'm stuck here. I know basically to let $x_0=0$ but I can't figure out (guess?) what $z_0$ should be. I set $g(y)=2x_0^2+3y^2-6z_0^2$ and so $g'(y)=6y$. I want to have $|g(y_0)|_p<|g'(y_0)|^2_p$, where $y_0$ is a root, so that I can use Hensel's Lemma again.
Case 3: $\mathbb Q_3$ (I haven't got round to this yet.)
Firstly, is this the correct method to approach this question? Secondly, is there some nice way to determine what $z_0$ should be or is it just a typical trial and error to see which $z_0$ works?