Question

(a) If f(x)and g(x) are associates in F[x], show that they have the same roots in F.

(b) If $\mathit{f}\left(x\right)\mathbf{,}\mathit{g}\left(x\right)\mathbf{\in }\mathit{F}\left[x\right]$have the same roots in F, are they associates in F[x] ?

Short answer

It is shown that,

1. f(x) and g(x) have the same roots in F.
2. No, f(x) and g(x) are not associates in F[x].

Step-by-step solution

a)Step 1: Determine f(x) and g(x) have same roots in F

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

Assume that, $f\left(x\right)=cg\left(x\right)$for non-zero constant $c\in F$.

Now let’s consider, $a\in F$be so that $f\left(a\right)=0$. Then,

$g\left(a\right)=c^{-1}f\left(a\right)=c^{-1}0=0$

Such that roots of f(x) are the root of g(x).

If g(x) = 0, then $f\left(a\right)=cg\left(a\right)=c0=0$, thus root of g(x) are root of f(x).

b)Step 2: Determine and are associate with

No, f(x) and g(x) are not associates inF[x].

Let’s consider that, $F=\mathrm{\mathbb{Q}}$and polynomials are:

$x-1and{\left(x-1\right)}^2=x^2-2x+1$

It has the same set of roots but it has different degrees, therefore, it cannot be an associate in F[x] .