An element $x$ of a Hopf algebra $H$, is called a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$. The set of primitive elements of $H$ is denoted $P(H)$. It can be shown that:
"If $H$ is a $\mathbb{k}$-Hopf algebra of finite $\mathbb{k}$-dimension and $\mathbb{k}$ is a field of characteristic zero, then $P(H)=\{0\}$"
The proof of the above fact, amounts to showing -inductively- that, if there is $0\neq x \in P(H)$ then $x^{n}$ (for all positive integers $n$) are linearly independent, which is a contradiction to the finite dimensionality of $H$. See for example at Hopf algebras - an introduction (Dascalescu-Nastasescu-Raianu), exercise 4.2.16.
Now the question: If we drop the requirement of characteristic zero (for the field $\mathbb{k}$ of coefficients) what are some examples violating the above proposition? In other words, considering hopf algebras over some finite field, are there -explicit- examples of primitive elements?