Diophantine equation generated by the maximal subfield of a circular field

Using the fundamental basis of the field $L_9=\mathbb{Q} (2\cos(\pi/9))$, the form $N_{L_9}(\gamma)=f(x, y, z)$ is found and the Diophantine equation $f(x,y,z)=a$ is solved. A similar scheme is used to construct the form $N_{L_7}(\gamma)=g(x,y,z)$. The Diophantine equation $g (x, y, z)=a$ is solved.