Show that $\mathbb{Q}(\sqrt{3},\sqrt[4]{3}, \sqrt[8]{3},...)$ is algebraic over $\mathbb{Q}$ but not a finite extension.

Question

Show that $\mathbb{Q}(\sqrt{3},\sqrt[4]{3}, \sqrt[8]{3},...)$ is algebraic over $\mathbb{Q}$ but not a finite extension.

I think for the algebraic part, since for every simple extension, each of those elements can be adjoined and each of these simple extensions has a minimal polynomial that cannot be reduced in $\mathbb{Q}$. For example, the simple extension $\mathbb{Q}(\sqrt{3}, \sqrt[4]{3})(\sqrt[8]{3})$ has minimal polynomial $x^{8}-3$. And since each simple extension has an increasingly large degree, the degree of the simple extensions over the previous extension gets larger for each attachment. But I am not sure how to express this formally...

For the infinite degree part, I was thinking because the set $\left \{\sqrt{3},\sqrt[4]{3}, \sqrt[8]{3},...\right \}$ is linearly independent?

Answer 1

Any element $\alpha$ of $F$ is a rational expression in the numbers adjoined. As such, it can involve only finitely many of the $\sqrt[2^k]3$. If in such an expression, $\sqrt[2^n]3$ is one with maximal $k$, then all other $\sqrt[2^k]3$ are powers of $\sqrt[2 k]3$. It follows that $\alpha\in\Bbb Q(\sqrt[2^n]3)$ and $\alpha $ is algebraic.

A different approach for the infinity part: By Eisenstein, the polynomial $X^{2^n}-3$ is irreducible. Hence $[F:\Bbb Q]\ge[\Bbb Q(\sqrt[2^n]3):\Bbb Q]\ge 2^n$, where $n$ is arbitrariy. It follows that $[F:\Bbb Q]$ is inifnite.

Answer 2

Let $L = \mathbb Q(3^{\frac 1{2^n}})$.

Why is $L$ algebraic over $\mathbb Q$? It is because the generating set of $L$ is $\mathbb Q \cup \{3^{\frac 1{2^n}}\}$, which are all contained in $\bar{\mathbb Q}$, the algebraic closure of $\mathbb Q$. Therefore, $L \subset \bar {\mathbb Q}$, which by definition of the algebraic closure implies it is algebraic over $\mathbb Q$.

However, $L$ contains subfields of the form $K_n$, where $K_n = \mathbb Q(3^{\frac 1{2^n}})$. One can conclude that $[K_n : \mathbb Q] = 2^n$, since $x^{2^n} - 3$ is irreducible via the Eisenstein criterion(shifted).

Then, suppose $L$ were finite, then $[L : \mathbb Q] < \infty $, but by the tower property it is equal to $[L : K_n][K_n : \mathbb Q]$ for each $n$. In particular, $[L : \mathbb Q]$ is a multiple of $2^n$ for all $n$, clearly impossible if it is finite.