Three things:

1. Unless I misunderstand, it is $A^{-\sigma-1}$ which is ambiguous.

2. In a similar way, in your case it is $A^{-\sigma^3-\sigma^2-\sigma-1}$.

3. For very similar questions you should look up the notion of Lagrange resolvent.

Sorry, my first answer was too hasty. However, I maintain that you should look up the notion of Lagrange resolvent, which can be found in many algebraic number theory books. In your case it is $A^{s^3+2s^2+3s+4}$ which is ambiguous.