Prove that two propositions are equivalent

Question

Let $K$ be a field with $16$ elements and $f \in K[X]$ a polynomial with coefficients in $K$. Prove that the following two propositions are equivalent:

$1$. There exists $g \in K[X]$ such that $f = g'$

$2$. There exists $h \in K[X]$ such that $f = h^2$

I haven't managed to do anything meaningful yet.

Thank you!

Answer 1

Hint

If $f=g'$, every term of $f$ has even degree.