Can we always find field extensions of a given number field and a given degree?

Solution 1:

For $p$ prime, once you know that $x^{p^k}−2\in \Bbb{Q}[x]$ is irreducible for all $k$ (or just that $\bigcup_k \Bbb{Q}(2^{1/p^k})$ is not a finite extension),

take the largest $k$ such that $x^{p^k}-2$ has a root $a$ in $F$. You'll get that $x^p−a\in F[x]$ is irreducible.

Proof: assuming it is reducible, with an irreducible factor $f(x)=\prod_{i=1}^j(x-a^{1/p}\zeta_p^{c_i})$ and $jl=1+mp$ you'll get that $(−1)^{jl} f(0)^l a^{-m}= \zeta_p^{l\sum_{i=1}^j c_i} a^{1/p}\in F$ is a root of $x^p-a$, contradicting that $k$ is the largest.

You've constructed an extension $F(a^{1/p})/F$ of degree $p$ arbitrary so you can construct a tower of extensions of degree $n$.