Can someone please explain what are the fields with characteristic 2 with examples?
In the book "Lectures in Abstract Algebra", in section 5, it is written that it is convenient to treat separately the symmetric products over fields of characteristic 2. But it does not explain why. Why is that true?
The reason for treating them differently is that the relationship to derived quadratic forms breaks down.
If you have a symmetric bilinear form $B$, you can immediately make a quadratic form by saying $Q(x)=B(x,x)$. When the characteristic is something other than $2$, you can take a quadratic form $Q$ and prove $B(x,y):= \frac12 (Q(x+y)-Q(x)-Q(y))$ is a symmetric bilinear form. When the characteristic is $2$, you can't invert $2$ to do that, so you have to take other measures.
For every power of $2$ and every positive integer $n$, there is a unique field of order $2^n$.
You could also take the field of rational polynomials $F(x)$ to get more characteristic $2$ fields.