Relationships of Eigenvalues in Algebraic Closure Suppose that $k$ is a field, and $A \in M_n(k)$  is a matrix that becomes diagonalizable over $\overline{k}$, the algebraic closure of $k$. Let $\lambda_1, \ldots, \lambda_n$ denote the (not necessarily distinct) eigenvalues of $A$ in $\overline{k}$.


*

*Must each $\lambda_i$ be in $k$?

*Must $\sum_i \lambda_i$ be in $k$?

*Must $\sum_i \lambda_i^2$ be in $k$?

*Must $\sum_{i<j} \lambda_i\lambda_j$ be in $k$?

*Must $\sum_{i<j} (\lambda_i - \lambda_j)$ be in $k$?


I think #1 is almost certainly false, otherwise the algebraic closure wouldn't be necessary. For #2, I believe that it is true, since the sum of the diagonal entries (all elements of $k$) is equal to the sum of the eigenvalues. I am not even sure how to start for the rest.

2., 3., 4. are symmetric polynomial in the eigenvalues, hence can be expressed using the elementary symmetric polynomials, which occur as coefficients in ...?


*

*and 5: Consider the $2\times 2$ real matrix $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$

