I am reading Papantonopoulou - Algebra, Ch 8, Rings of Polynomials
Theorem 8.2.2 (Division Algorithm) reads as follows: (see attachment for full proof)
Let F be a field and f(x) and g(x) elements of F[x], with g(x) 0 . Then there exist unique elements Q(x) and r(x) of F[x] such that
(1) f(x) = q(x)g(x) + r(x)
(2) r(x) = 0 or deg r(x) deg g(x)
I am having trouble following the uniqueness argument which runs as follows:
Suppose we have:
where or deg deg g(x).
Subtracting the two equations give we get
We must have , for otherwise deg deg g(x) which would makje equation (3) impossible .... etc etc
(see the proof on the attachment)
Can anyone show me (formally) why the last statement ie " , for otherwise deg deg g(x) which would makje equation (3) impossible " follows?