Endomorphism ring of a quiver representation

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Consider the quiver with one loop, and consider the representation on $\mathbb{F}_{2}(t)$ described by the matrix $$W=\begin{pmatrix} 1 & t \\ 1 & 0 \end{pmatrix}.$$ I need to calculate the endomorphism ring of this representation. I was able to calculate this on $\mathbb{F}_{2}(t)$, but now I need to calculate it one The algebraic closure of $\mathbb{F}_{2}(t)$. I'm stuck.