Suppose p0,…,pm∈P(F) are such that pj has degree j. Prove that p0,…,pm is a basis of Pm(F).
p0 is a constant, it's nonzero since if it were zero it would have degree −∞ (see 2.12).
Our strategy will be to write the standard basis in terms of p0,…,pm then apply 2.42.
We can write 1=p0/p0, now consider how p1=ax+b for some a,b∈F with a=0. We can write x=(p1−b)/a. Continue like this to write xk as a linear combination of of p0,…,pk. Thus span(p0,…,pm)=Pm(F) and so p0,…,pm is a basis by 2.42.