Question

Let ,$f\left(x\right),g\left(x\right),h\left(x\right)\in F\left[x\right]$ with$f\left(x\right)$ and$g\left(x\right)$ relatively prime. If$h\left(x\right)\overline{)f\left(x\right)}$ , prove that$h\left(x\right)$ and $g\left(x\right)$are relatively prime.

Short answer

It is proved that h(x) and g(x) are relatively prime.

Step-by-step solution

Statement of theorem 4.8 

Theorem 4.8states that $F$is a field and $a\left(x\right),b\left(x\right)\in F\left[x\right]$ , both non-zero. Then, there exists a uniquegreatest common divisor$d\left(x\right)$of$a\left(x\right)$ and$b\left(x\right)$ . Also, there exist polynomials (not unique) $u\left(x\right)$and $v\left(x\right)$in which $d\left(x\right)=a\left(x\right)u\left(x\right)+b\left(x\right)v\left(x\right)$.

Show that hx and gx  are relatively prime

According to theorem 4.8, there exist $u\left(x\right),v\left(x\right)\in F\left(x\right)$in which $1=f\left(x\right)u\left(x\right)+g\left(x\right)v\left(x\right)$. There are certain $r\left(x\right)\in F\left[x\right]$in which $f\left(x\right)=h\left(x\right)r\left(x\right)$because $h\left(x\right)\overline{)f\left(x\right)}$. Then,

$\begin{array}{rcl}1& =& f\left(x\right)u\left(x\right)+g\left(x\right)v\left(x\right)\\ & =& h\left(x\right)\left(r\left(x\right)u\left(x\right)+g\left(x\right)v\left(x\right)\right)\end{array}$

According to the proof of theorem 4.8, $\left(h\left(x\right),g\left(x\right)\right)$would be a monic polynomial of the lowest degree which is a linear combination of $h\left(x\right)$and$g\left(x\right)$ .

As a result of the preceding calculation,$\left(h\left(x\right),g\left(x\right)\right)=1$ and thus,$h\left(x\right)$ and$g\left(x\right)$ are relatively prime.

Hence, it is proved that $h\left(x\right)$and$g\left(x\right)$ are relatively prime.