If $f(t,x)\in F_q(t)[x]$ is a Morse function, does this mean splitting field of $f(t,x)$ over $F_q(t)$ is a regular extension?

Question

One of the classical result of Hilbert says, if $f$ is a Morse function, then the splitting field of $f(X,T)$ over $Q(T)$ is a regular extension with Galois group $S_n.$ J. P.Serre- Topics in Galois theory, Theorem 4.4.1

Does the same hold for $f\in \F_q(T)[X]$, function field with finite elements and $q=p^n, n>0$ with positive characteristic "p"?

i.e., if $f(t,x)\in F_q(T)[X]$ is a Morse function, then does it mean that the splitting field of $f(X,T)$ over $F_q(T)$ is a regular extension?

Thank you.