Let x be transcendent over $\mathbb{F}$, where $\mathbb{F}$ is a field with characteristic 2. I have a polynomial $f=t^3+xt+x\in K[t]:=\mathbb{F(x)}[t]$. Let L be the splitting field of $f$. I know that $f$ is irreducible by Eisenstein, so I know that the degree $[L:K]\in\{3,6\}$. In a larger proof I am working on I need that $[L:K]=6$ but I have no idea how to prove it. Thanks in advance.
4kiraL