Question 4
Let $p \ge 2$ be a prime, $(K, +, \cdot)$ be a field with $p^2$ elements and
$$
L = \{0, 1, \dots, \underbrace{1+1+\cdots+1}_{p-1 \text{ times}}\}.
$$
Prove that for every $x \in K \setminus L$ there exists $a, b \in L$ so that $x^{2013} + ax + b = 0$.
$$
L = \{0, 1, \dots, \underbrace{1+1+\cdots+1}_{p-1 \text{ times}}\}.
$$
Prove that for every $x \in K \setminus L$ there exists $a, b \in L$ so that $x^{2013} + ax + b = 0$.