Question

Prove Corollary 4.9.

Short answer

Corollary 4.9 is proved.

Step-by-step solution

Statement of corollary 4.9

Corollary 4.9considers $F$ as a field and$a\left(x\right),b\left(x\right)\in F\left[x\right]$ , both non-zero. Thegreatest common divisorof $a\left(x\right)$ and $b\left(x\right)$ such that $d\left(x\right)$ satisfies the following conditions:

1. $d\left(x\right)\overline{)a\left(x\right)}$ and $d\left(x\right)\overline{)b\left(x\right)}$
2. When $c\left(x\right)\overline{)a\left(x\right)}$ and$c\left(x\right)\overline{)b\left(x\right)}$ , then$c\left(x\right)\overline{)d\left(x\right)}$ .

Show that corollary 4.9

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 $d\left(x\right)=a\left(x\right)u\left(x\right)+b\left(x\right)v\left(x\right)$.

Theorem 4.7states that $F$ is a field and $a\left(x\right),b\left(x\right)\in F\left[x\right]$, with $b\left(x\right)$ non-zero.

1. When $b\left(x\right)$divides$a\left(x\right)$ , then $cb\left(x\right)$ divides $a\left(x\right)$ with some non-zero $c\in F$.
2. All divisors of $a\left(x\right)$ contain degrees less than or equal to$dega\left(x\right)$ .

Assume that$d\left(x\right)=\left(a\left(x\right),b\left(x\right)\right)$ . From the definition of gcd, $d\left(x\right)\overline{)a\left(x\right)}$and$d\left(x\right)\overline{)b\left(x\right)}$ therefore, condition (i) is satisfied.

Assume that $c\left(x\right)\overline{)a\left(x\right)}$ and $c\left(x\right)\overline{)b\left(x\right)}$, as a result, there are $r\left(x\right),s\left(x\right)\in F\left[x\right]$ in which $a\left(x\right)=c\left(x\right)r\left(x\right)$ and $b\left(x\right)=c\left(x\right)s\left(x\right)$. According to theorem 4.8, there are $u\left(x\right),v\left(x\right)\in F\left[x\right]$ such that

$\begin{array}{rcl}d\left(x\right)& =& a\left(x\right)u\left(x\right)+b\left(x\right)v\left(x\right)\\ & =& \left[c\left(x\right)r\left(x\right)\right]u\left(x\right)+\left[c\left(x\right)s\left(x\right)\right]v\left(x\right)\\ & =& c\left(x\right)\left[r\left(x\right)u\left(x\right)+s\left(x\right)v\left(x\right)\right]\end{array}$

As a result, $c\left(x\right)\overline{)d\left(x\right)}$, condition (ii) is satisfied.

Now, assume that $d\left(x\right)$ satisfies conditions (i) and (ii). From condition (i), $d\left(x\right)$ would be a common divisor of $a\left(x\right)$and$b\left(x\right)$ , and from condition (ii) and theorem 4.7.

When $c\left(x\right)$ would be a common divisor of $a\left(x\right)$ and$b\left(x\right)$ then, $degc\left(x\right)\le degd\left(x\right)$.

As a result, $d\left(x\right)=\left(a\left(x\right),b\left(x\right)\right)$.

Hence, corollary 4.9 is proved.