Question

Without using statement (2), prove directly that statement (l) is equivalent to statement (3) in Theorem 4.12.

Short answer

p(x) is irreducible when its divisor is associate and a non-zero constant polynomials.

Step-by-step solution

To prove

The main objective is to prove that without using statement (2), prove directly that statement (l) is equivalent to statement (3) in Theorem 4.12.

Applying Theorem 4.12

Consider that p(x) is irreducible then, its divisor is associate and non-zero constant polynomials.

Let’s consider r(x) and s(x). If both are associate of p(x) so that p(x)=ap(x)²for a$\in$F, it is a contradiction by unique factorization. Thus, either r(x) or s(x) is a non-zero constant polynomial.

Now, assume that p(x) for all r(x) and f(x) so that as it has either r(x) or s(x) as non-zero constant polynomial. Generally, without loss, let’s say that r(x)=r$\in F$.

Then,$P\left(x\right)=r\left(x\right)$ , therefore, p(x) and s(x) are associate.

Therefore, all divisors of p(x) are either non-zero constant polynomials or its associate, that means p(x) isirreducible.