I'm trying to find something out about an algebraically closed field $k$ with char $k \notin \{2,3\}$. (Specifically, the number of irreducible $k$-linear representations of $S_4$.) Now I no little of ring theory, and nothing about field theory. My guess however is that I can treat $k$ as if it were $\mathbb{C}$, as follows:
The divisors of $S_4$ are 2, 3, 4, 6, 8, 12, 24. As the number of irreducible representations equals the number of conjugacy classes, and the number of conjugacy classes equals the number of simple $k[G]$-modules (up to isomorphism), by Maschke's Theorem, we are looking for $k[G]$-modules for which char $k \nmid \#G = 24$. Now (and here's where I'm bluffing) the other divisors (besides 2 and 3) are excluded because they are not prime, so that only leaves char $k = 0$, so $k \cong \mathbb{C}$...?
Edit
In my proof I make use of the set of class functions $$\mathbb{C}_{\text{class}}(S_4) = \{f : S_4 \rightarrow \mathbb{C} \mid (\forall \sigma, \tau \in S_4)(f(\sigma \tau \sigma^{-1}) = f(\tau))\}.$$
Would it be enough to generalise this to $k_{\text{class}}(S_4)$ for the given $k$. If so, how would one go about this?