SCHWEITZER 1969 Problem 2

Let $p\geq 7$ be a prime number, $\zeta$ a primitive $p$ th root of unity, $c$ a rational number. Prove that in the additive group generated by the numbers $1,\zeta,\zeta^2,\zeta^3+\zeta^{-3}$ there are only finitely many elements whose norm is equal to $c$ . (The norm is in the $p$ th cyclotomic field.)

K. Gyory