Is there a good way to see what the units of the group ring $\mathbb{F}\_p[\mathbb{F}\_p]$ (p is a prime) are? 



Description of the units of the group ring Fp[Fp] ?
You can use that $\mathbb{F}\_p[\mathbb{Z}/p]\cong \mathbb{F}[\epsilon]/\epsilon^p,$ where $\epsilon=1-\sigma$ for $\sigma$ a generator of the group $\mathbb{Z}/p$. This is an Artin local ring, and from this you can get in any number of ways that units are $R\setminus I$ where $R$ is your ring and $I$ is the ideal spanned by $\epsilon$. 


