Wedderburn Decomposition of Finite Group Rings Over Algebraically Closed Fields

Dummit and Foote Abstract Algebra, section 18.2:

Let $G$ be a finite group, $F$ a field, and $FG ≅ R = R_1 × ... × R_r$ be the Wedderburn decomposition of the group ring $FG$. Each $R_i$ is a ring of $n_i × n_i$ matrices with entries from a division ring $Δ_i$. We show in this case that in fact $Δ_i ≅ F$.

We have seen already that $Z(R_i)$ is the set of scalar $n_i × n_i$ matrices with entries from $Z(Δ_i)$. Since $F⊆Z(FG)$, this means (after composition with the projection map $π_i : R → R_i$) $F$ embeds into $Z(Δ_i)$ for each $i$. Now, consider the left ideal $I_i$ of $R$ consisting of elements with $0$ in every coordinate but the $i^\text{th}$, where the value may be any $n_i × 1$ column matrix. This left ideal $I_i$ is a left $R$-module, with an action that restricts to turn $I_i$ into a $Δ_i$-module (multiplication with scalar matrices in $R_i$), which in turns restricts to turn $I_i$ into a vector space over $F$. As a left $FG$-module (inherited from isomorphism of $FG$ with $R$), we may decompose $I_i$ into a finite direct sum of cyclic $FG$-modules (of rank $≤n_i$), hence $I_i$ is a finite vector space over $F$ (of dimension $≤n_i|G|$). $I_i$ is also a ring, with a subring $J_i$ (of $1×1$ matrices) isomorphic to $Δ_i$. Since $J_i$ is an $F$-subspace of $I_i$, it too is finite dimensional, with a vector space action agreeing with multiplication from $F⊆J_i$. That is to say, ultimately, that $Δ_i$ is a finite dimensional vector space over $F$ such that $F⊆Z(Δ_i)$.

If $F$ is algebraically closed, then necessarily $F=Δ_i$, since each ring extension $F[α]⊆Δ_i$ of $F$ by an element $α∈Δ_i$ will in fact be a field extension by $F⊆Z(Δ_i)$, hence be of degree one as a vector space over $F$ by algebraic closure, implying $α∈F$.