I agree that there should be an internal nth root method, 


There is. It is x.nth_root(n).


Now the PR https://github.com/sagemath/sage/pull/38362 adding deprecation 
warnings to the use of AA(x)^(1/n) to get the real root is ready.

After one year deprecation period, the 
PR https://github.com/sagemath/sage/pull/38564 will change the behavior of 
AA(x)^(1/n) as promised.

Please look around both PRs. If there is no objection for #38362, I will 
set it to positive review soon. 

If anyone has objections, please state them in #38362. 


 

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