Question

Verify that Theorem 4.23 is valid with$\mathrm{\mathbb{Z}}$ and$\mathrm{\mathbb{Q}}$ is replaced by R and F.

Short answer

It is verified that Theorem 4.23 is valid with$\mathrm{\mathbb{Z}}$ and$\mathrm{\mathbb{Q}}$ replaced by R and F.

Step-by-step solution

Theorem 4.23

Let$f\left(x\right)$ be a polynomial with integer coefficients. Then,$f\left(x\right)$ factor as a product of polynomial of degree m and n in$\mathrm{\mathbb{Q}}\left[x\right]$ if and only if$f\left(x\right)$ factors as a product of polynomials of degree m and n in$\mathrm{\mathbb{Z}}\left[x\right]$ .

Prove the Converse Part

Let $f\left(x\right)\in R\left[x\right]$.

Since R is UFD $\Rightarrow R\left[x\right]$is also UFD.

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

Hence, $f\left(x\right)$can be written as product of polynomial $g\left(x\right)h\left(x\right)$in $R\left[x\right]$.

Now, by Corollary 10.37,$f\left(x\right)$ is reducible in$F\left[x\right]$ can be written as a product of polynomial$g\left(x\right)h\left(x\right)$ in$F\left[x\right]$ .

Prove Implies Part

By Exercise 6 of Section 10.5, If $\mathit{f}\left(x\right)$is primitive in $R\left[x\right]$and irreducible in $F\left[x\right]$then, localid="1653728909123" $\mathit{f}\left(x\right)$is irreducible in $R\left[x\right]$.

Let $f\left(x\right)\in F\left[x\right]$and $f\left(x\right)$is irreducible in $F\left[x\right]$.

By Corollary 10.34,it is a product of two primitive polynomials.

Therefore, $f\left(x\right)=g\left(x\right)h\left(x\right)$.

Now, we know that an irreducible polynomial of positive degree is primitive.

Therefore, $f\left(x\right)$is primitive and irreducible in $F\left[x\right]$.

Hence, by Exercise 6 of section 10.5 $f\left(x\right)$ is irreducible in $R\left[x\right]$.

Therefore, it can be written as a product of polynomials of positive degree as: $f\left(x\right)=g\left(x\right)h\left(x\right)$in $R\left[x\right]$.

Thus, it is verified that Theorem 4.23 is valid with$\mathrm{\mathbb{Z}}$ and$\mathrm{\mathbb{Q}}$ replaced by R and F.