Question

Prove that f(x) and g(x)are associate in F[x]if and only if f(x)|g(x) andg(x).|f(x)

Short answer

Hence, it is proved that f(x) and g(x) are associates in F[x].

Step-by-step solution

Monic associate

It is given that the polynomials are $f\left(x\right),g\left(x\right)\in F\left[x\right]$.

Steps for Monic associate

From exercise 4.2.4(a), it is implies that if f(x)|g(x) and g(x)|f(x) then, f(x) and g(x) are associates in $F\left[x\right]$.

Conversely, let $f\left(x\right),g\left(x\right)\in F\left[x\right]$ and if $f\left(x\right)=cg\left(x\right)$ where c is a non-zero element, $c\in F\left[x\right]$ then f(x)=cg(x) implies $g\left(x\right)=c^{-1}f\left(x\right)$.

Therefore, from equations $f\left(x\right)=cg\left(x\right)$ and $g\left(x\right)=c^{-1}f\left(x\right)$ it gives f(x)g(x)| and g(x)|f(x).

Hence, f(x) and g(x) are associates in F[x].