Assume  That $U,V$ are two  filters on the natural  number $\mathbb{N}$.
We say that $U$ is  equivalent to $V$ if there  is a  bijection $\phi: \mathbb{N} \to  \mathbb{N}$  such that $\tilde{\phi}(U)=V$ where $\tilde{\phi}:P(\mathbb{N}) \to  P(\mathbb{N})$ is  the  natural extension of  $\phi$ to the  power set $P(\mathbb{N})$.
Let $U,V$ be two  non principal ultra filter on $\mathbb{N}$.

Let $\mathbb{R}^*_{U}$ and $\mathbb{R}^*_{V}$ be the corresponding [nonstandard extension of real numbers](https://en.wikipedia.org/wiki/Non-standard_analysis#Basic_definitions) associated with $U$ and $V$, respectively.

Assume that  $\mathbb{R}^*_{U}$ and $\mathbb{R}^*_{V}$  are isomorphic as  fields. Does  this  imply that $U$ and  $V$ are  equivalent filters?

My  apology in advance, if the  question is  elementary. The  question arose me about  17 years  ago when I was  trying  to  understand the  application of  non standard  analysis  to ordinary  differential equations.