I came across an example in Stewart's Galois Theory wherein we have field $K$ and a simple extension $K\subset K(t)$.
In an example we are asked to find the degree $[\mathbb{Z}_5(t):\mathbb{Z}_5]$.
Initially I said okay $\mathbb{Z}_5(t)=\{a+ bt:a,b\in \mathbb{Z}_5\}$, and so a basis for the vector space $\mathbb{Z}_5(t)$ over $\mathbb{Z}_5$ is $\{1,t\}$ thus the degree $[\mathbb{Z}_5(t):\mathbb{Z}_5]$ is 2. However the answer is "infinite". I then thought that the set describing this field will be a generic combination of powers of $t$?
In another example, we are given the case when $K=\mathbb{Z}_2$ and are asked to describe the subfields of $K(t)$ of various forms, e.g. $K(t^2)$ and $K(t+1)$. I'm not even sure where to start with this!
To summarise my queries: Is there anything we can say about a generic extension $K\subset K(t)$? How do you describe the field whenever t is an unknown? How do we obtain that the degree is infinite? How can we go about describing the subfields of such a field (without applying actual Galois theory methods - these are introductory field theory exercises).
Thanks