I would like to understand the definition of a simple transcendental extension and the theorem that states all such extensions are isomorphic. So for example, if $K \subseteq \mathbb{C} $ is any subfield and $L:K$ a field extension with $\alpha \in L $ then $K(\alpha) \cong K(X)$, where $K(X)$ is the field of rational functions in the indeterminate $X$. First, what makes a field extension transcendental? And second, what is the explicit isomorphism between $K(\alpha)$ and $K(X)$? I've heard this isomorphism called the "evaluation homomorphism" so could you please explain what this map is doing specifically?
My understanding of this theorem is that in this context $\alpha$ (like $X$) is purely a mathematical symbol, and is no longer considered as the root of some polynomial over $K$. Thus, it is as if we are writing rational expressions in $X$ as those in $\alpha$. However, this interpretation seems banal to me (why even write it in $\alpha$ if we have it in $X$?) and what exactly makes $K(X)$ transcendental in the first place?