Valuations and different ideal in local fields We know that if we have a totally ramified finite extension of local fields $L/K$ then the different ideal is  $D_{L/K} = ( h'(\pi))$ i.e. the ideal generated by te derivative of the minimal polinomial of $\pi$ (a uniformizer of L). My question is: since the uniformizer is not unique then if we consider another one uniformizer $\tilde{\pi}$ what happens to $D_{L/K}$ and its valuation is still the same? (i.e. the valuation of the minimal polynomial ?)
Thank you for the answers!

Another uniformizer has the form $\tilde{\pi} = u \pi$ for some unit $u \in O_L^{\times}$. The minimal polynomial $\tilde h$ is just $\tilde h(X) = h(u^{-1}X)$, and in particular we get
$$\tilde h'(\tilde{\pi}) = u^{-1} h'(u^{-1} \tilde{\pi}) = u^{-1} h(\pi).$$
Thus the ideal generated by $\tilde h'(\tilde{\pi})$ in $O_L$ is the same as the ideal generated by $h'({\pi})$.
