I quote Proposition 2.3, page 14 lines -3 and -4 of Michael Rosen's book


Number Theory in Function Fields:


Let $b\_n$ be the number of square-free monics in $A= \mathbb{F}\_q[t]$ of degree $n.$
Then $b\_1=q$ and for $n>1$, $b\_n=q^n-q^{n-1}.$


Using the proposition it is easy to prove that any polynomial $P \in A$ can be written
$$
P = S\_1+S\_2
$$
where $S\_1$ and $S\_2$ are square-free, provided 


$$
q \neq 2
$$


Question: What happens when $q=2.$



All polynomials over a finite field are sums of $2$ square-free polynomials
Alright, putting together all the comments it appears to still be true when $q = 2$, depending on how you like your square-free polynomials.


First, to quote Sonia's comment, If $P$ is not sq.free, then for each sq.free polynomial $Q$ of degree $n$, we have a non-zero $P−Q$, of degree $\leq n−1$, $2^{n−1}$ in all. The number of sq.free polynomials of degree $\leq n−1$ is $2^{n−1}$. So there must be nonempty intersection. – Sonia Balagopalan


Now, suppose $P$ is square-free and has degree $n$. For any square-free polynomial of lesser degree, $Q$, $P-Q$ is a polynomial of degree $n$. There are $2 + 2 + 2^4 + \ldots + 2^{n-2} = 2^{n-1}$ choices for such $Q$, but there are only $2^{n-1}$ non-square-free polynomials of degree $n$ and we certainly did not land on $0$, so at least one of these must be another square-free polynomial of degree $n$.


So any polynomial of degree $n \geq 3$ can be written as the sum of two square free polynomials of degrees $n$ and $d$ where $d < n$. What about when $n = 2$? The only polynomials to consider are


$P\_1 = t^2,$


$P\_2 = t^2 + 1,$


$P\_3 = t^2 + t$ and 


$P\_4 = t^2 + t + 1$.


If you want to add the restriction that our two square free polynomials MUST have degree at most 2, as Sonia claims, $P\_3$ and $P\_4$ are counterexamples. Otherwise we may write


$P\_1 = (t^2 + t) + (t)$,


$P\_2 = (t^2 + t) + (t + 1)$,


$P\_3 = (t^2 + t^7 + 1) + (t + t^7 + 1)$ and


$P\_3 = (t^2 + t^7 + t) + (t^7 + 1)$.


(Note: There is nothing too special in picking the $7$.)


Upon further thought, you can do what you want with the degree 1 polynomials. A similar trick using a larger degree polynomials will work, or you can use those as counterexamples to a restricted degree version if you like. Although trivial, you may also want to express $0$ and $1$ as sums, which can certainly be done as $z$ and $z+1$ are both square-free.


