I was working on a problem and it got reduced to finding all maximal ideals of $KS_3$ where $K$ is a field. (I'm particularly interested in two cases, when $\text{char}K\in \{2,3\}$ and $\text{char}K\not \in \{2,3\}$)
Is there any way I can do this with basic knowledge? (without character theory, or Artin-Wedderburn theorem, ...)