Wei Yue asks the following question: In the definition of a homomorphism between two formal groups and , what is the reason we required ? The short reason is that we want to preserve the origin, that is, . A more complicated reason is that in general, the expression may not make sense for a power series if has a non-zero constant term. This is a point we have been somewhat cavalier about: When can we substitute one power series in another and get a well-defined power series as the answer? You should ponder this for yourself a bit, and then try to formulate some conditions precisely using the definition
of the ring of power series with coefficients in .
Here is an exercise worth trying out: Suppose is a polynomial that defines an algebraic group law on the field with as the origin. Then is the only choice.
Hang on, what about
Well, you have to formulate a bit carefully what is meant by an algebraic group law. Anyways, the conclusion is that deforming the usual group structure in any reasonable sense requires us to move out of the realm of polynomials.