Let $k$ be an algebraically closed field of characteristic $p>0$. Let $L$ be the extension
of $k(t)$ obtained by attaching a root of an irreducible polynomial $f\in k(t)[x]$.
Is there a way to tell from the form of $f$ when the extension $L/k(t)$ is unramified at all
the finite places of $k(t)$? (i.e. the places generated by $t-a$ for some $a\in k$).


Are there criteria for when this is not the case,
for example does $f$ have to involve only powers of $x$ divisible by $p$ except for the linear and constant term?



Which polynomials define extensions of $k(t)$ unramified at the finite places
Presumably you know that $f$ is separable over $k(t)$ or else you wouldn't pose the question. Scale $x$ by $k(t)^{\times}$ so that $f$ becomes $x$-monic in $k[x,t]$ with $df/dx \ne 0$. Now the irreducible plane curve $C := \{f=0\}$ inside the $(x,t)$-plane over $k$ has projection to the affine $t$-line that is *finite* flat and generically etale (as $f$ is $k(t)$-separable), with non-etale locus supported exactly over the zeros of the $x$-resultant $R \in k[t]$ of $f$ and $df/dx$ (with the resultant nonzero since $f$ is $k(t)$-separable). 


Thus, in the "generic" case that $C$ is *smooth* it follows that $k[C]$ is the integral closure of $k[t]$ in $k(C)$, so the absence of ramification in $k(C)$ over finite places of $k(t)$ is exactly the condition that the resultant $R = {\rm{Res}}\_x(f, df/dx) \in k[t] - \{0\}$ is constant. This is an extremely difficult condition to arrange (apart from cases where $df/dx$ doesn't involve $x$ at all, such as $f = x^{p^n} - x + h(t)$ corresponding to an Artin-Schreier covering).


The smoothness of $C$ can be determined algorithmically from $f$ (without having to compute actual points on $C$) by applying the effective Nullstellensatz to test if 1 lies in the ideal $(f,df/dx,df/dt)$ in $k[x,t]$, and the $x$-resultant $R \in k[t]$ can be computed very directly from $f$ as well. But perhaps this is not really a condition on "the form of $f$" (whatever that may mean).


Aside from Artin-Schreier covers, it is a very hard problem to build finite etale covers of the affine line by purely algebraic methods without some serious geometric machinery. In general there's probably no way to detect by simple means from staring at "the form of $f$" and avoiding genuine calculations if it gives an example (apart from $f$ arising by the Artin-Schreier construction).


