Working with function field extensions

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Let $K=k(t)$, where $k$ is a finite field. Consider a rational function $F(t)\in K$ and a simple finite extension $L=K(u)$. For instance, take $L=k(u,t)$, where $u^5=t$.

My first question is: How do we evaluate $F(u)$?

The following code I am using produces an error (... NotImplementedError)

k.<a>=GF(19^2)
Rk=k['t']
K.<t>=Frac(Rk)
Rx.<x>=PolynomialRing(K)
L.<u>=K.extension(Rx(x^5-t),'u')
F=(t^25+t^5)/(t^5+1)
F(u)
F.subs(t=u)

Notice that in my example $F(u)=(t^5+t)/(t+1)$ is a function of $t$, since $u^5=t$.

My second question is: How would I coerce $F(u)=g(t)$ back into $k(t)$?