$\dim_{\mathbb F_p}(\mathbb F_p[[T]]/(T^l))=l$

Question

Let $\mathbb F_p[[T]]$ be the ring of formal series over $\mathbb F_p$, and $l\in\mathbb N.$

How to prove that: $\dim_{\mathbb F_p}(\mathbb F_p[[T]]/(T^l))=l$ ?

Answer 1

It has basis $(1,T,T^2,\dots,T^{l-1})$. Observe that $T^l=0$ in the quotient ring, so all higher powers vanish, too.