Question 25
Let $p$ be a prime, $p \geq 3$, and $k$ an odd number, not a multiple of $p$. Let $K$ be a finite field with $kp+1$ elements and $A = \{x_1, x_2, \dots, x_t\}$, the set of elements $K^* = K \setminus \{0\}$ of order different of $k$ in the multiplicative group $(K^*, \cdot)$. Prove that the polynomial $P(X) = (X + x_1)(X + x_2)\dots(X + x_t)$ has at least $p$ coefficients equal to 1.