What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2? As pointed in the comments, it can be reduced to $X^2+X+c=0$ with $c\neq 0$.
Usual completion of square breaks. I know For a method (different from checking all elements) for finite fields but field there is Chen Formula that roughly looks like $X=\sum_{m} c^{4^m}$. I need it for any am more interested in the local field $F((z))$ or actually an arbitrary field of characteristic 2.
