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?
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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 $\epsilon$. 

