A generalization of the Euler Phi function

Let R be a Dedekind domain and f\in R a nonzero element. Then (f) admits a unique factorization into prime ideals (f)=P_1^{e_1}\cdots P_k^{e_k}. Our definition for the generalization of the Euler Phi function will rely on this decomposition.

Definition 1: In the situation above, we define \phi_R(f)=\# R/(f)^\times to be the Phi function of R.

This phi function satisfies a multiplicative property much as the original does. To see this, let P,Q be two distinct nonzero prime ideals. Since P,Q are nonzero prime ideals of a Dedekind domain, they are both maximal, and we have P+Q=R. Further, we have r(P^e+Q^f)=r(r(P^e)+r(Q^f))=r(P+Q)=r(R)=R. By the Chinese remainder theorem this gives an isomorphism R/P^eQ^f\cong R/P^e\times R/Q^f and thus (R/P^eQ^f)^\times \cong (R/P^e\times R/Q^f)^\times. Putting this all together shows to give a complete calculation of \phi_R it suffices to give a calculation of \#R/P^e for every prime ideal P and every power e. The feasibility of this task is probably hopeless in general. However, for integers of the fields \mathbb{Q} and \mathbb{F}_q(t) there is a complete description of this function.

Proposition 2: (The Euler Phi) We have the following formulas for \mathbb{Z}:

  1. \phi_\mathbb{Z}(p)= p-1 when p is prime
  2. \phi_\mathbb{Z}(p^m)=p^m-p^{m-1} for any m\geq 1 and p prime
  3. \phi_\mathbb{Z}(n)=n\prod\limits_{p\mid n} (1-\frac{1}{p}).

Proposition 3: Similarly, we have the following formulas for \mathbb{F}_q[t]:

  1. \phi_{\mathbb{F}_q[t]}(f)=q^d-1 when f is irreducible of degree d
  2. \phi_{\mathbb{F}_q[t]}(f^m)=q^{md}-q^{md-d} for any m\geq 1 and f irreducible of degree d
  3. \phi_{\mathbb{F}_q[t]}(g)=q^{\text{deg}(g)}\prod\limits_{f\mid g} (1-\frac{1}{q^\text{deg}(f)}).

The proof for proposition 2 is pretty standard in elementary number theory texts and the proof for proposition 3 follows essentially these same arguments so they are omitted. In fact, I’m including them so I can remember this problem in the future (and it took me longer than I would like to admit to show property 2 of proposition 3 – and no effort should be wasted). In the future, if I find formulas for the Phi function of other Dedekind domains, I will include them here.

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