This is [crossposted](https://math.stackexchange.com/questions/522826/wedderburn-decomposition-of-d-5) from MSE. The question:



> 
> Find the Wedderburn decomposition of $D\_{5},$ the dihedral group of order 10, over the ﬁeld $\mathbb{F}\_{3}.$
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> 
> 


I have shown that the irreducible representations of $D\_{5}$ over $\mathbb{F}\_{3}$ are the two trivial 1-dimensional representations, and two 4-dimensional representations which derive from considering the action of $D\_{5}$ on the group of $5^{\text{th}}$ roots of unity in $\mathbb{F}\_{3}.$


Now, we can decompose the group ring $\mathbb{F}\_{3}(D\_{5})$ into a direct sum of two copies of $\mathbb{F}\_{3}$ and two copies of $\mathbb{F}\_{3}^{4}$. However, it seems that the two copies of $\mathbb{F}\_{3}^{4}$ combine to form the matrix ring $M\_{2\times 2}(\mathbb{F}\_{3^{2}})$, giving the final Wedderburn decomposition into matrix rings. 


Is there a nice explanation for this last step? I would appreciate any help in understanding this.



Wedderburn decomposition of $D_{5}$
In the following paper the authors deal with the Wedderburn decomposition of group algebras of finite metacyclic groups over a finite field:


G.K. Bakshi - S. Gupta - I.B. Passi: Semisimple metacyclic group algebras, Proc. Indian Acad. Sci., Math. Sci. 121, No. 4, 379-396 (2011).


It is available at this link:


<http://www.ias.ac.in/mathsci/vol121/nov2011/pmsc-d-10-00210.pdf>


The Wedderburn decomposition you are looking for is discussed as an example in Section 5 of the mentioned paper.


