Diophantine Equation Generated by the Maximal Subfield of a Circular Field

Abstract

© 2020, Allerton Press, Inc. 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.