Question

If f(x) is primitive in R[x] and irreducible in F[x], prove that f(x) is irreducible in R[x].

Short answer

It is proved that$f\left(x\right)$ is irreducible in$R\left[x\right]$ .

Step-by-step solution

Take contradiction

Let $f\left(x\right)$be not irreducible in $R\left[x\right]$.

Then, role="math" localid="1653722808400" $f\left(x\right)=g\left(x\right)h\left(x\right)$for some $g\left(x\right),h\left(x\right)\in R\left[x\right]$.

Now,$f\left(x\right)$ is primitive in$R\left[x\right]$ .

Use Theorem 10.34

By Theorem 10.34, we can say that, $g\left(x\right),h\left(x\right)$are also primitive in $R\left[x\right]$.

Since $g\left(x\right)$and $h\left(x\right)$are polynomials of positive degree in $F\left[x\right]$, this contradicts the irreducibility of $f\left(x\right)$. $\left[\because f\left(x\right)isirreducibleinF\left[x\right]\right]$

Hence, $f\left(x\right)$is irreducible in$R\left[x\right]$ .