(Context: polynomial multiplication using DFT/FFT)
Let $f = \sum\limits_{i=0}^{n-1} f_i x^i$ and $g = \sum\limits_{j=0}^{n-1} g_j x^j$ be polynomials in $F[x]$ for some field $F.$ The convolution of $f$ and $g$ is given by $f \ast g = \sum\limits_{k=0}^{n-1} c_k x^k$ where $c_k = \sum\limits_{i+j \equiv k \bmod n} f_i g_j.$ The $k$th coefficient in the product $fg$ is given by $\sum\limits_{i+j = k} f_i g_j.$ Why is convolution equivalent to multiplication in $F[x]/(x^n-1)?$ Shouldn't that be mod $x^n$?