Let $G$ be a finite group with normal subgroup $N$. Let $\chi$ be an irreducible complex character of $G$. Then Clifford's Theorem says that $\mathrm{res}^{G}_{H}\chi = e(\eta_1 + \cdots + \eta_r)$$\mathrm{res}^{G}_{N}\chi = e(\eta_1 + \cdots + \eta_r)$ where $e$ is a positive integer and each $\eta_i$ is a complex irreducible character of $N$ such that the $\eta_i$'s are the distinct conjugates of $\eta := \eta_1$ (i.e. for each $i$ there exists $g_i \in G$ such that $\eta_i(n)=\eta(g_i n g_i^{-1})$ for all $n \in N$.) Let $\mathbb{Q}(\eta) = \mathbb{Q}(\eta(n) : n \in N)$ be the character field of $\eta$ and similarly for $\mathbb{Q}(\mathrm{res}^{G}_{N} \chi)$. Since $\mathbb{Q}(\eta_i)=\mathbb{Q}(\eta)$ for each $i$, we have $K:=\mathbb{Q}(\mathrm{res}^{G}_{N} \chi) \subseteq \mathbb{Q}(\eta)$.
Question: Is $e$ somehow related to $[\mathbb{Q}(\eta):K]$? For example, do we have $e \geq [\mathbb{Q}(\eta):K]$? Otherwise, can you tell me anything about the relation between $K$ and $\mathbb{Q}(\eta)$? e.g. do we somehow get $\mathbb{Q}(\eta)$ from $K$ by adjoining certain roots of unity? If so, which ones (are they related to say $[G:N]$)? (I am actually interested in the case where we replace $\mathbb{Q}$ by the $p$-adics $\mathbb{Q}_p$ throughout and $[G:N]$ is a power of $p$.)