Solution
= Solution
The left side is the norm form $N_{K/\mathbb Q}(x+y\alpha+z\alpha^2)$ for $\alpha^3=7$. Reducing a hypothetical equation modulo 7 and then descending through the forced divisibilities of $x,y,z,t$ gives an infinite descent, so only the zero solution exists. If $N(\theta)=3$, clearing denominators minimally and applying the norm equation would produce a forbidden nonzero integer solution; hence no such $\theta$ exists.