gcdhx g x 1F and gcdhx fx px then gcdhx g xfx px 12B Prove T

gcd(h(x), g (x) 1F and gcd(h(x), f(x) p(x), then gcd(h(x), g (x)f(x)) p(x). 12B) (Prove Theorem 4.10) Let F be a field and let fo), g(x) and h(x) be polynomials in Flxl. Prove that if gcd(g(x), h(x) iF and g (x) divides f(x)h(x), then g(x) divides f(s).

Solution

12A. Given h(x) ,g(x) are relatively prime   

gcd[ h(x) , f(x) ] = p(x) => h(x) = p(x) q(x) , f( x) = p(x) r(x) where q, r are relatively prime

consider the gcd of [ h(x) , g(x) f(x) ] = gcd of [ p(x) q(x) , g(x) p(x) r(x) ]

=p(x) { as the only common factor is p(x) , as h,g are relatively prime , no common factor between q and g ]

b . g(x) divides f(x) h(x) means

g(x) divedes only f(x) as there is no common factor between g and h