Question

Prove that any two nonzero polynomials in${\mathbf{\mathbb{Q}}}_{\mathbf{\mathbb{Z}}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}$ have a gcd.

Short answer

It has been proved that any two nonzero polynomials in${\mathrm{\mathbb{Q}}}_{\mathrm{\mathbb{Z}}}\left[\mathrm{x}\right]$ have a gcd.

Step-by-step solution

Find decomposition factors

Let $f\left(x\right)\in {\mathbb{Q}}_{\mathbb{Z}}\left[x\right]$ be a non-zero polynomial.

By exercise 34, it can be written as$f\left(x\right)=c{x}^n{p}_1{\left(x\right)}^{n^1}\mathrm{...}{p}_k{\left(x\right)}^{n^k}$ where$c\in \mathbb{Q}$ , $n\ge 0$.

Also, this decomposition is unique upto the order of the factors.

Find gcd

Now, if $g\left(x\right)=c\text{'}{x}^n{p}_1{\left(x\right)}^{m^1}\mathrm{...}{p}_k{\left(x\right)}^{m^k}$is factored similarly then let $d=\min \{n,m\}$and$d_i=\min \{n_i,m_i\}$ .

Therefore, gcd is$(c,c\text{'})x^d{p}_1{\left(x\right)}^{d_1}\mathrm{...}{p}_k{\left(x\right)}^{d_k}$ .

Conclusion

It can be concluded that any two nonzero polynomials in${\mathbb{Q}}_{\mathbb{Z}}\left[x\right]$ have a gcd.