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 $\overline{k}$.


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*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.

Hint: the characteristic polynomial of a matrix, $\chi_A(x) := \det(xI-A)$, has as roots the eigenvalues of $A$.  Can you express your sums in terms of this polynomial's coefficients?
(Think in terms of symmetric polynomials.  If your sums are not symmetric, you should look for counter-examples, perhaps with $k=\mathbb{R}$.)
Further notes: If $\lambda_1,\ldots,\lambda_k$ are the eigenvalues of $A$, then
$$\chi_A(x)= (x-\lambda_1)\cdots (x-\lambda_k).$$
The constant coefficient is then $\pm \lambda_1 \cdots \lambda_k$, and the $x$-coefficient is $$\pm\bigg((\lambda_2 \cdots \lambda_k)+(\lambda_1\lambda_3 \cdots \lambda_k)+(\lambda_1\lambda_2 \lambda_4 \cdots \lambda_k) + \cdots + (\lambda_1 \lambda_2 \cdots \lambda_{k-1}\bigg)$$
(It's easiest to ignore the exact signs.)  There are general formulas for these coefficients in terms of the roots of our polynomial, called the elementary symmetric polynomials.  It is a fundamental result that any symmetric polynomial (that is, a polynomial that remains unchanged by any permutation of it's inputs) can be written as a sum of symmetric polynomials.
For your problem, you'll want to use that the value of any symmetric polynomial lies in the field generated by the values of your elementary symmetric polynomials.  But the elementary symmetric polynomials show up as the coefficients of $\chi_A$, so they lie in the base field.
