Rigorous proof of the argument that one cannot square $-1$ in complex number problems

I would like to ask a fundamental question with regards to the imaginary number and it is something many beginners are told are wrong, but I would like to seek a rigorous proof of why it is wrong. It is a question I faced when my student asked me this. Take for example, $(-1)^{1/6}$. We can compute this in 2 different ways:

$$(-1)^{1/6}=[(-1)^{2}]^{1/12}=1$$

or
$$(-1)^{1/6}=[(-1)^{1/3}]^{1/2}=(-1)^{1/2}=i$$
I understand both the methods above are wrong, and the typical response is that you cannot square $-1$. Is there a rigorous proof as to why this method is flawed? Perhaps using abstract algebra or Galoise Theory?

Thank you!