Does ramification have anything to do with inseparability? It feels like an extension of Q in which p ramifies should somehow correspond to an extension of F_p(t). Does totally ramified <--> purely inseparable?

In fact, saying an irreducible polynomial f(x) is inseparable is the same as saying that f(x) ramifies when we extend Q[x] to L[x], where L is the splitting field of f(x).

By correspond, I generally mean taking an extension of Q defined by a root of p(x)=r, where r is a rational making the extension nontrivial, and then extending F_p(t) by a root of p(x)=t. It's interesting because then this extension of F_p(t) corresponds to a number of extensions of Q (this is the same thing when you do Galois theory by looking at fundamental groups of branched coverings of C. Then do you look at etale fundamental groups of objects associated to these function fields over finite fields?).

equivalentto saying that the extension $L[x]/K[x]$ of Dedekind domains is unramified - that is, irreducible polynomials in $K[x]$ do not have any repeated factors in $L[x]$. – David Corwin Aug 23 '10 at 23:50