Is there any results about the Euler's totient function for ideals?
More precisely, for any number field $k,$ if we define $\phi(\mathfrak{a})$ as the number of residue classes $\bar{\mathfrak{a}}\in\mathfrak{o}_k/\mathfrak{a}$ such that gcd$(\bar{\mathfrak{a}},\mathfrak{a})=\mathfrak o_k$. Then is the fuction $\phi(\mathfrak{a})$ sharing the basic properties that the usual Euler 's totient function has?
For example, for all $\alpha\in\mathfrak{o}_k$ prime to $\mathfrak{a}$ we have $\alpha^{\phi(\mathfrak{a})}\equiv1\mod\mathfrak{a};$ for any prime ideal $\mathfrak{p}$ and for any $\alpha\in\mathfrak{o}_k$ $\alpha^{\phi(\mathfrak{p})}\equiv\alpha\mod\mathfrak{p}$ and $\phi(\mathfrak{a})=N(\mathfrak{a})\prod_{\mathfrak{p}|\mathfrak{a}}\left(1-\frac{1}{N\mathfrak{p}}\right).$
Any comment will be acknowledged!