RO-NMO 2006 Problem 121

Let $\mathcal K$ be a finite field. Prove that the following statements are equivalent:

(a) $1+1=0$ ;

(b) for all $f \in \mathcal K \left[ X \right]$ with $\textrm{deg} f \geq 1$ , $f \left( X^2 \right)$ is reducible.