Proof that the number $\sqrt[3]{2}$ is irrational using Fermat's Last Theorem

In this comment BCnrd argues that this proof is "essentially circular", because converting an FLT counterexample to a Frey curve with certain congruence conditions as in the Wiles proof requires an argument equivalent to establishing irrationality of $\sqrt[3]{2}$.

In case not, I doubt anyone on Earth knows. The Wiles' proof is a huge document (150 pages), readable by only a few people, which indirectly involves the work of dozen (hundredths) mathematicians, thousand (million ?) pages of previous results. Unless you've read all this corpus, you can't tell whether the irrationality of $\sqrt[3]2$ is somewhere invoked or not.

---

In any case, there is no circular argument as the irrationality of $\sqrt[3]2$ can be established by a child of five.

Your argument is correct but there is no need to use Wiles' proof of Fermat's Last Theorem: an elementary proof of the case $n=3$ was given by Euler.