Simplification of a sum with roots of unity

Question

Let $p$ be an odd prime, $\zeta $ a primitive $p-$th root of unity and $${a_n}(x) = \sum\limits_{k = 1}^{p - 1} {\prod\limits_{j = 1}^n {\left( {1 + {\zeta ^{jk}}x} \right)} } .$$ It seems that for $0 \leqslant i < p$ $${a_{pn + i}}(x) = {b_i}(x){\left( {1 + {x^p}} \right)^n}$$ for some polynomial $b_i(x)$ of degree $i$.

Such things are probably well known. I would like to know where I can find references.

Answer 1

For fixed $k$, the product of $(1+\zeta^{jk}x)$ over $p$ consecutive values of $j$ equals $1+x^p$. Your claim follows.