Is there any "famous" family of trinomials over finite fields? 
For example over $F\_2$ we have 


$$
f(x)=x^{2\times 3^k}+x^k+1
$$



Family of irreducible trinomials over finite fields
I don't know if it is famous, but there is a family due to Ore:


$x^q -x - a$ is irreducible over $F\_q$ whenever $a \in F\_q^\times.$ This is a result of O. Ore, 


*Ore, O.*, [**Contributions to the theory of finite fields.**](http://dx.doi.org/10.2307/1989836), Transactions A. M. S. 36, 243-274 (1934). [ZBL60.0111.04](https://zbmath.org/?q=an:60.0111.04).


