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 $\mathrm{res}^{G}\_{N}\chi = e(\eta\_1 + \cdots + \eta\_r)$ 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 $p$.)



Character fields and Clifford's theorem
I don't think there's any clear cut relationship. In particular, the inequality $e \ge [\mathbb Q(\eta) : K]$ needn't hold. For example, take $G=S\_5$, $N=A\_5$ and let $\chi$ be the unique irreducible $S\_5$-character of degree $6$. Then we have $\mathrm{res}^{G}\_{N} \chi = \eta + \bar{\eta}$, where $\eta$ and $\bar{\eta}$ are the degree $3$ irreducible characters of $A\_5$. So here $e=1$ and $\mathbb Q(\eta) = \mathbb Q(\sqrt 5)$. On the other hand, $\chi$ is integer valued so $K=\mathbb Q$. Thus $e < [\mathbb Q(\eta) : K]$ in this case.


