What are the automorphisms of the 2-variable polynomial ring over $\mathbb{F}_2$, the field with 2 elements? Are they generated by $(x \mapsto y, y \mapsto x)$, $(x\mapsto x+ p(y), y\mapsto y)$, and $(x \mapsto x, y \mapsto y + p(x))$ where $p$ runs over all polynomials over $\mathbb{F}_2$? These are automorphisms, right?
I can see that any automorphism must fix the constants, but not much more.