Question 65
Let $p$ be a prime number, $n$ a positive integer not divisible by $p$, and $\mathbb{K}$ a field with $p^n$ elements, with unit element $1_{\mathbb{K}}$ and zero element $\hat{0} = 0_{\mathbb{K}}$. For any $m \in \mathbb{N}^*$ we denote $\hat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \dots + 1_{\mathbb{K}}}_{m \text{ times}}$ and we define the polynomial
$$
f_m = \sum_{k=0}^{m} (-1)^{m-k} \widehat{C_m^k} X^{p^k} \in \mathbb{K}[X].
$$
a) Show that the set of the roots of the polynomial $f_1$ is $\{\hat{k} \mid k \in \{0, 1, \dots, p-1\}\}$.
b) Let $m \in \mathbb{N}^*$ be arbitrary. Determine the set of the roots in the field $\mathbb{K}$ of the polynomial $f_m$.
$$
f_m = \sum_{k=0}^{m} (-1)^{m-k} \widehat{C_m^k} X^{p^k} \in \mathbb{K}[X].
$$
a) Show that the set of the roots of the polynomial $f_1$ is $\{\hat{k} \mid k \in \{0, 1, \dots, p-1\}\}$.
b) Let $m \in \mathbb{N}^*$ be arbitrary. Determine the set of the roots in the field $\mathbb{K}$ of the polynomial $f_m$.