Suppose Fermat's Little theorem is generalized to $q$-th cyclotomic polynomials (for $q$ prime) in the following manner: $$(x+a)^n=x^n+a = x^{(n \bmod q)}+a \pmod {\Phi_q(x),n}$$
if $n$ is prime.
Let $\zeta_q$ be a primitive $q$-th root of unity, and $R$ the ring of integers in the $q$-th cyclotomic field $\mathbb Q(\zeta_q)$. Let $d ∈ R$ be a cyclotomic integer, and $P(x)$ (or just $P$) be the minimal polynomial of $d$.
Is there a simple identity (similar to Fermat's Little Theorem) for $$(x+a)^n \pmod {P,n}$$ assuming $n$ is prime?
For example, choosing $q=5$, we have an identity (like Fermat's Little Theorem) assuming $n$ is prime: $$(x+a)^n = x^{(n \bmod 5)}+a \pmod {x^4+x^3+x^2+x+1,n}.$$
Given the field $\mathbb Q(\zeta_5)$ and cyclotomic integer $d=\zeta_5^3 - \zeta_5$, $P=x^4-5x+5$ is the minimal polynomial of $d$. What is the identity following from Fermat's Little Theorem for: $$(x+a)^n \pmod {x^4-5x+5,n} = $$ if $n$ is prime?
The answer would involve some kind of arithmetic in cyclotomic fields, but it doesn't appear that the modular congruences are random if $n$ is prime (should be for most composite $n$, however).