Suppose $T: V(F) \rightarrow V(F)$ is a linear transformation, where $V$ is a finite dimensional vector space. What is the necessary and/or sufficient condition that $F$ contains all eigenvalues of the matrix (corresponding to the linear transformation $T$)?
Note: A is a real symmetric matrix, all its eigenvalues are real.
For rational symmetric matrices, are there any such classifications? Means, if $A \in M_{n\times n}(\mathbb{Q})$ over the field $\mathbb{Q}$, then under what conditions $\lambda(A) \subset Q?$