Suppose that $\sqrt[3]{2}=\frac{p}{q}$. Then $2q^3 = p^3$ i.e $q^3 + q^3 = p^3$, which is contradiction with Fermat's Last Theorem.
My question is whether this argument is a correct mathematical proof, since Fermat's Last Theorem is proven, or does it loop on itself somewhere along the proof of the Theorem?
In other words, does the proof of Fermat's Theorem somehow rely on the fact that $\sqrt[3]{2}$ is irrational?
UPD:
As pointed out in comments, this actually is a valid argument, no matter what was used in the proof of the Fermat's Last Theorem (which from now on will be referred to as the Proof). What really interests me, is whether the Proof uses on some step the fact that $\sqrt[3]{2}$ is irrational?
Best Answer
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 $\sqrt[3]{2}$.