Character fields and fields of definition

Let $G$ be finite and let $\chi$ be the character of a complex representation of $G$. Its character field is $\mathbb{Q}(\chi ):=\mathbb{Q}(\{\chi (g):g\in G\})\subseteq \mathbb{C}\mathrm{.}$ More generally, if $F\subseteq \mathbb{C}$, write $F(\chi )$ for the field generated by $F$ and the values of $\chi$ in the sense of Field extensions, generated subrings $F[S]$, generated subfields $F(S)$, and simple extensions.

If $F\subseteq \mathbb{C}$, a complex representation $\rho :G\to \mathrm{GL}(V)$ is realizable over $F$, and $F$ is a field of definition for $\rho$, when there is a finite-dimensional representation ${\rho }_F:G\to \mathrm{GL}(W)$ over $F$ such that $\mathbb{C}{\otimes }_{F}W$ is equivalent to $V$. Thus a character field records traces, whereas a field of definition records a matrix model; neither term asserts that the other field has the other property.