Suppose we adjoin a symbol $k$ to the field $F_p$, as in $F_p(k)$. What is an intuitive understanding of the structure of this field and its elements? Since multiplication needs to be closed, all new polynomials are part of this field. But also multiplicative inverses need to be included, does that mean all rational functions over $F_p$ should also be included? What about additive inverses? Do you have a mental model of this structure that gives you a sense of what the elements look like and how they preserve the structure, or is that something you can only reasonably inspect case-by-case?
Let's consider $F_2(k)$. If my understanding above is correct, all the following are new elements added to $F_2$:
The list goes on forever, which doesn't seem right. Perhaps the structure is not well-defined if $k$ is just some arbitrary symbol, and it needs to be a root of a polynomial that doesn't have a root in $F_2$. This would mean I can only make $F_p(k)$ a field if it is an algebraic extension of $F_2$. Is that so?
If it is, then say I choose $p(x)=x^2+x+1$ in $F_2$, and I add the root $k$. How do I define $k^n$ for any power $n$, and how do I define $k+k+k$. In other words, how do I force the field structure? $k$ being the root of $p(x)$ should somehow help me fill in the multiplication and addition tables, but I don't see how.