If $\alpha$ is transcendental over $F$, we sometimes say that $[F(\alpha): F] =\infty$. I'm wondering if this is true in a literal sense, namely: is there some irreducible polynomial of "infinite" degree which has $\alpha$ as one of its roots?
My first attempt at formalizing this question was:
(*) Can we find some sequence of polynomials $f_1, f_2,\dots$ with roots $r_1, r_2,\dots$ such that $\lim_{n\to\infty} r_n =\alpha$?
Unfortunately, at least in some cases the answer to this question is "yes" for an uninteresting reason: if we have some Cauchy sequence $q_1, q_2,\dots$ which converges to $\alpha$, then $\{(x - q_i)\}_i$ meets (*) but isn't describing a polynomial of "infinite" degree in any meaningful sense.
So I'm wondering: is there some meaningful sense in which polynomials of infinite degree can have roots that polynomials of finite degree do not? And if so, can all transcendental numbers the written as the root of such polynomials?
Answer
I'm not sure if it is entirely within the spirit of your question, but this road you are walking will eventually lead you to the idea of classifying transcendental numbers by how well they can be approximated by "simple" algebraic numbers. Take a look into stuff like Mahler's Classification.
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