In which field (besides $\mathbb{R}$) is every symmetric matrix potentially diagonalizable?

In which field (besides the well-known $\mathbb{R}$) is every symmetric matrix potentially diagonalizable? A matrix is potentially diagonalizable in a field $F$ if it is diagonalizable in the algebraic closure of $F$.

It appears to me that the fields $\mathbb{F}_2$ and $\mathbb{C}$ do not have this property. What about other finite fields?

from Hot Weekly Questions - Mathematics Stack Exchange