Question

Prove Corollary 4.13.

Short answer

The corollary 4.13 is proved by theorem 4.12.

Step-by-step solution

To prove Corollary 4.13

This is proved by induction on n. For n=1,this statement is trivial.

Now, let’s assume that it is true for products of up to n-1 polynomials. And assume $p\left(x\right)|\prod _{i-1}^n{a}_i\left(x\right)$.

Using Theorem 4.12

Applying theorem 4.12, it can be either $p\left(x\right)|\prod _{i-1}^n{a}_i\left(x\right)$or $p\left(x\right)|a_n\left(x\right)$.

In the latter case, it is done but in the former, the inductive hypothesis gives $p\left(x\right)|a_i\left(x\right)$one of the $1\le i<n$.

Hence, it is proved.