How to prove that Reed Solomon codes are cyclic

Question

Suppose $C$ is the $[n,k]_q$ Reed-Solomon code with $n = q-1$ and evaluation points defined as $\alpha_i = \beta^{i-1}$ (for $ i = 1, \dots, n$) where $\beta$ is a generator of multiplicative group $F^{\star}_{q}$. How to prove that $C$ is a cyclic code?

Answer 1

Hint: If evaluating the polynomial $f(x)$ at the points $\alpha_i$ gives the codeword $(\gamma_1,\gamma_2,\ldots,\gamma_n)$, what do you get when you evaluate the polynomial $f(\beta x)$? Observe that $f(x)$ and $f(\beta x)$ have the same degree and that $\beta^n=\beta^0$.