- #1
swampwiz
- 571
- 83
I don't understand proof of uniqueness theorem for polynomial factorization, as described in Stewart's "Galois Theory", Theorem 3.16, p. 38.
"For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the factors are written."
"Suppose that f = f1 ... fr = g1 ... gs where f is a polynomial over K and f1 ... fr , g1 ... gs are irreducible polynomials over K. If all the fi are constant then ... so are all the gj are constant."
So far so good.
"Otherwise, we may assume that no fi is constant by dividing out all the constant terms."
How can this assumption be made? What if some of the fi are constant, and some are not constant?
There is more unclear text here. Does anyone have a link to better explanation of this?
"For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the factors are written."
"Suppose that f = f1 ... fr = g1 ... gs where f is a polynomial over K and f1 ... fr , g1 ... gs are irreducible polynomials over K. If all the fi are constant then ... so are all the gj are constant."
So far so good.
"Otherwise, we may assume that no fi is constant by dividing out all the constant terms."
How can this assumption be made? What if some of the fi are constant, and some are not constant?
There is more unclear text here. Does anyone have a link to better explanation of this?