In caracteristic $p$, there is a finite subgroup $ \alpha_p \subset \mathbb{G}_a $ where we define the functor of points of $ \alpha_p $ by associating to a $ k $ - algebra $ R $ : $$ \alpha_p (R) = \{ \ c \in \mathbb{G}_a (R) \ | \ c^p = 0 \ \} $$ This is represented by the scheme $ \mathrm{Spec} k[t] / (t^p ) $ and so $ \alpha_p $ is unipotent group.
Could you explain to me please, in details, ans step by step, how do we reason to show that $ \alpha_p $ is not smooth ?
Thanks in advance for your help.