For an algebraic field extension $[L:K]$ with $l\in L$ and $p_{_l}\in K[X]$ the minimal polynomial of $l$, is there a basis independent proof that $K[X]/p_{_l}\cong K(l)$?
I know a proof that uses $\{x_i+(p_{_l})\}_{i<deg(p_{_l})}$ as a basis for $K[X]/p_{_l}$ and $\{l^i\}_{i<deg(p_{_l})}$ as a basis for $K(l)$, but is there a standard basis free proof? (perhaps using that both fields coequalize the same two arrows in the category of fields?)
Hint: What is the Kernel of the canonical ring homomorphism $$K[X] \to K(l),$$ $$X \mapsto l?$$
It would also be instructive to consider what happens if $K \subset L$ is an arbitrary extension of fields and $l \in L$ is transcendental over $K$.