The residue field of a prime ideal of a cyclotomic number field Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th  root of unity in $\mathbb{C}$.
Let $K = \mathbb{Q}(\zeta)$.
Let $A$ be the ring of algebraic integers in $K$.
Let $p \ne l$ be a prime number.
Let $f$ be the order of $p$ modulo $l$, i.e. the smallest positive integer such that $p^f \equiv 1$ (mod $p$).
Let $P$ be a prime ideal of $A$ lying over $p$.
My question: Is the following proposition true? If yes, how would you prove this?
Proposition
Let $\alpha \in A$.
Then there exist rational integers $a_0, ..., a_{f-1}$ such that
$\alpha \equiv a_0 + a_1\zeta + ... + a_{f-1}\zeta^{f-1}$ (mod $P$).
Here $a_0, ..., a_{f-1}$ are uniquely determined mod $p$.

By my answer to this question, the degree of $P$ is $f$.
Let $\omega$ be the image of $\zeta$ by the canonical homomorphism $\mathbb{Z}[\zeta] \rightarrow \mathbb{Z}[\zeta]/P$.
Clearly $\mathbb{Z}[\zeta]/P = (\mathbb{Z}/p\mathbb{Z})(\omega)$.
Hence every element $x \in \mathbb{Z}[\zeta]/P$ can be uniquely written as $x = a_0 + a_1\omega + \cdots + a_{f-1} \omega^{f-1}$, where $a_i \in \mathbb{Z}/p\mathbb{Z}$.
This completes the proof.
