Question

Is the given polynomial irreducible?

(a) ${\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{3}\mathbf{}\mathbf{in}\mathbf{}\mathbf{\mathbb{Q}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{?}\mathbf{In}\mathbf{}\mathbf{ℝ}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{?}$

(b) ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{\mathbb{Z}}}_{\mathbf{3}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{?}\mathbf{}\mathbf{In}\mathbf{}{\mathbf{\mathbb{Z}}}_{\mathbf{7}}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{?}$

Short answer

$x^2-3$is irreducible in $\mathbb{Q}\left[x\right]$ and reducible in $ℝ\left[x\right]$

$x^2+x-2$is reducible in ${\mathbb{Z}}_3\left[x\right]$ and${\mathbb{Z}}_7\left[x\right]$

Step-by-step solution

Irreducible element

It is given that the polynomial is $x^2-3$ and it has to show irreducible in $\mathbb{Q}\left[x\right]$

and in $ℝ\left[x\right]$

If roots of a given polynomial lie in the given field, the polynomial is reducible; otherwise, irreducible.

Step 2: Check the polynomial irreducible or not

(a) Check for the polynomial $x^2-3$

$\begin{array}{c}p\left(x\right)=x^2-3\\ x^2-3=0\\ x^2=3\\ x=\pm \sqrt{3}\end{array}$

Therefore, $\pm \sqrt{3}\notin \mathbb{Q}\left[x\right]$but $\pm \sqrt{3}\in ℝ\left[x\right]$

Hence, the given polynomial is irreducible in $\mathbb{Q}\left[x\right]$and reducible in role="math" $ℝ\left[x\right]$

(b) Check for the polynomial $x^2+x-2$

$\begin{array}{c}p\left(x\right)=x^2+x-2\\ x^2+x-2=0\\ x^2+2x-x-2=0\\ x(x-1)+2(x-1)=0\\ (x-1)(x+2)=0\\ x=1,-2\end{array}$

Therefore, $1\in {\mathbb{Z}}_3\left[x\right]$and $1\in {\mathbb{Z}}_7\left[x\right]$

Hence, the given polynomial is reducible in ${\mathbb{Z}}_3\left[x\right]$and${\mathbb{Z}}_7\left[x\right]$