I am reading Papantonopoulou - Algebra - Ch 8 Rings of Polymonials
On page 251 (see attachment) Papantonopoulou states the Unique Factorization Theorem as follows:
Let F be a field and a non-constant polynomial. Then
(1) , where , and each is a monic irreducible polynomial over F
(2) Except for the order of the irreducible facotrs, the factorization in (1) is unique
The proof of (1) is fine, but I am having trouble with the start of the uniqueness proof which reads as follows: (see attachment)
Use induction on deg f(x) = n.
If n=1, uniqueness holds.
So now assume uniqueness holds for all polynomials of degree < n = deg f(x).
where u,v and the and are monic and irreducible, then
... etc etc
Can someone show me (formally) why follows in the argument above