[1] We choose these conditions so as to ensure that they reduce modulo the maximal ideal to Frobenius and the derivative at the origin is the prime element.
For each element a in Zp there is a unique endomorphism f of the Lubin–Tate formal group law such that f(x) = ax + higher-degree terms.
[2] We outline here a formal group equivalent of the Frobenius element, which is of great importance in class field theory,[3] generating the maximal unramified extension as the image of the reciprocity map.
Taking X and Y to be in the unique maximal ideal gives us a convergent power series and in this case we define F(X,Y) = X +F Y and we have a genuine group law.
This works by defining a family of modules (indexed by the natural numbers) over the ring of integers consisting of what can be considered as roots of the power series repeatedly composed with itself.