Are the exponential and matrix exponential valid over an arbitrary field $\mathbb{F}$?

Question

I understand that if $[A] \in M_{n\times n}(\mathbb{C})$ then the matrix exponential of $[A]$ is denoted $e^{[A]} \in M_{n\times n}(\mathbb{C})$, and is defined by $$e^{[A]} := \sum_{k=0}^\infty \frac{1}{k!} [A]^k$$ and that this is a generalization of $e^x: \mathbb{C} \to \mathbb{C}$. I wanted to know if we can say that $e^x: \mathbb{F} \to \mathbb{F}$ for some field $\mathbb{F}$, and if the matrix exponential would then be well-defined.