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In a previous question, I asked how to compute Brauer characters of the alternating group $\mathfrak{A}_3$; the answer to this question provided a solution for all cyclic groups.

I would now like to compute Brauer characters for some low-rank symmetric groups. As a toy example, let's start with $\mathfrak{S}_3$ when $p=2$.

My main problem once again is choosing the correct valuations. General theory tells us our characteristic-zero discrete valuation ring needs to contain all the cube roots of unity, so my guess is to take $K=\mathbb{Q}_2(\omega)$. We then need a valuation which extends the 2-adic valuation to give us a valuation ring whose residue field has characteristic 2 and contains all the cube roots of unity (I suppose it must be isomorphic to $\mathbb{F}_4$?) This is the part I'm stuck on.

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