Question

Prove (1)$\Rightarrow$(2) in Theorem 4.12

Short answer

Thus, the given statement is proved.

Step-by-step solution

To prove

The objective is to prove$\left(1\right)\Rightarrow \left(2\right)$ in theorem 4.12.

Determine theorem 4.12

If p(x) is irreducible, and p(x)|b(x)c(x).

Now, consider (p(x),b(x)),

p(x) is monic divisor of the irreducible p(x). Thus, it is either 1 or an associate of p(x).

If$\left(p\left(x\right),b\left(x\right)\right)=fp\left(x\right)$for $f\in F$then, p(x)|b(x) .

If$\left(P\left(x\right),b\left(x\right)\right)=1$ then, by theorem 4.10 it is proved that p(x)|b(x).

Hence, $\left(1\right)\Rightarrow \left(2\right)$is proved.