Let $K=\mathbb F_{673}(t)$, and consider the polynomial $f=t^3x^{2019}-t^4+2\in K[x]$.
I want to determine whether or not $f$ is separable.
So far, I've noticed that we can write $f$ as $(tx^{673})^{3}-t^4+2$, but I'm not sure what to do next.
2026-04-01 03:40:58.1775014858
How do I check whether a polynomial is separable over a function field?
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Hint: Observe that $$f'=2019\,t^3x^{2018}=673(3t^3x^{2018})=0,$$ so what is $\gcd(f,f')$?