Let be a Dedekind domain and a nonzero element. Then admits a unique factorization into prime ideals . Our definition for the generalization of the Euler Phi function will rely on this decomposition.
Definition 1: In the situation above, we define to be the Phi function of .
This phi function satisfies a multiplicative property much as the original does. To see this, let be two distinct nonzero prime ideals. Since are nonzero prime ideals of a Dedekind domain, they are both maximal, and we have . Further, we have . By the Chinese remainder theorem this gives an isomorphism and thus . Putting this all together shows to give a complete calculation of it suffices to give a calculation of for every prime ideal and every power . The feasibility of this task is probably hopeless in general. However, for integers of the fields and there is a complete description of this function.
Proposition 2: (The Euler Phi) We have the following formulas for :
- when is prime
- for any and prime
Proposition 3: Similarly, we have the following formulas for :
- when is irreducible of degree
- for any and irreducible of degree
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.