Showing that the quadratic form $Q(x,y,z,t)=x^2+y^2+z^2-7\cdot t^2$ is anisotropic on $\Bbb Q^4$

Question

I'm looking for help in order to find a prove that the quadratic form $Q(x,y,z,t)=x^2+y^2+z^2-7\cdot t^2$ on $\mathbb Q^4$ can or cannot take the value $0$ on a nonzero element of $\Bbb Q^4$.

I was thinking that it is impossible because $7$ isn't the sum of three square of integers, but I don't see how to prove it...

Thanks

Answer 1

There is a rational solution if and only if there is an integer solution if and only if there is an integer solution for which $\gcd(x,y,z,t) = 1$.

When dealing with squares, looking at things modulo $8$ is often very useful. What can you say about the solutions to

$$ x^2 + y^2 + z^2 + t^2 \equiv 0 \pmod 8 $$

? (I've used $-7 \equiv 1$ to simplify the equation)