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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!

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You're letting $\phi(\mathfrak a)$ be the size $(\mathfrak o_k/\mathfrak a)^*$. The first two properties are then basic facts about groups, right? – Dylan Moreland Jun 13 '12 at 12:38
    
Yes, I see it, @Dylan, then the last one is obvious. – Qiang Zhang Jun 13 '12 at 12:50
    
Hi, do you have any references for this problem? I heard there was some research on it. – Francis Begbie Jan 19 at 3:49

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