Infinite algebraic extension of $\mathbb{Q}$

Question

I have this problem in a exercise list:

"Prove that $K=\mathbb{Q}(2^{\frac{1}{2}},2^{\frac{1}{3}}, 2^{\frac{1}{4}}, \ldots)$ is an algebraic extension, but not a finite extension of $\mathbb{Q}$."

What I did:

Let $\alpha \in K$, then $\alpha = a_0 + a_12^{\frac{1}{n_1}}+a_22^{\frac{1}{n_2}}+\cdots+a_m2^{\frac{1}{n_m}}$.

Now, since each factor of the sum above is algebraic over $\mathbb{Q}$, it follows that $\alpha$ is indeed algebraic over $\mathbb{Q}$ (because the set of algebraic numbers is a field).

Suppose now that $K$ is a finite extension of $\mathbb{Q}$. Then, by Steinitz's theorem, there is $u \in K$ such that $K=\mathbb{Q}(u)$. Let $p(x)$ be the minimal polynomial of $u$, and let $n$ be $p$'s degree. Then, K is an extension os degree $n$. However, the minimal polynomial of $2^{\frac{1}{n+1}}$ has degree $n+1$, so K is an extension of degree at least $n+1$, contradicting the hypothesis that $K$ is an extension of degree $n$. So, $K$ is an infinite extension of $\mathbb{Q}$.

My doubts: Can $\alpha$ be an infinite sum? If yes, can I use the same argument?

Answer 1

As has been mentioned, the result is a direct consequence of the fact that $[\mathbf{Q}(2^{1/n}):\mathbf{Q}]$ = n. This is because we then have, for every $n$, the inequality $$n = [\mathbf{Q}(2^{1/n}):\mathbf{Q}] \leq [K:\mathbf{Q}].$$ Since $[K:\mathbf{Q}]$ is larger than every natural number, it must be infinite.

The trickiest point is why $2^{1/n}$ has degree $n$ over $\mathbf{Q}$. For this it is enough to establish that the polynomial $X^n - 2$ is irreducible. This follows immediately from Eisenstein's criterion for the prime $p = 2$.

There is an error in your proof that every element of $K$ is algebraic over $\mathbf{Q}$. It is not clear at all why you claim any $\alpha \in K$ can be written in the form of a linear combination of various roots of $2$. The only thing that is obvious is that $\alpha$ can be written as a polynomial expression of these numbers, not a linear combination.

The proof that $K$ is algebraic is as follows. The field $K$ is generated by the set $S = \mathbf{Q} \cup \{2^{1/2}, 2^{1/3}, \dots\} \subseteq \overline{\mathbf{Q}}$, where $\overline{\mathbf{Q}}$ is the field of algebraic numbers over $\mathbf{Q}$. It follows that $K$ itself is contained in $\overline{\mathbf{Q}}$. Thus $K$ is algebraic over $\mathbf{Q}$.