Let $p$ be a prime and $\mathbb{Q}_p$ denotes the $p$-adic numbers. Is it true that the degree of the nontrivial $\mathbb{Q}_p$-irreducible representations of a cyclic group of order $p^n$ is divisible by $p-1$ ?

Proof or reference are appreciated.

Let $p$ be a prime and $\mathbb{Q}_p$ denotes the $p$-adic numbers. Is it true that the degree of the nontrivial $\mathbb{Q}_p$-irreducible representations of a cyclic group of order $p^n$ is divisible by $p-1$ ?

Proofs or references are appreciated.