Question

Express$x_4-4$ as a product of irreducible in $\mathrm{\mathbb{Q}}\left[x\right]$, in role="math" localid="1648646593814" $\mathrm{ℝ}\left[x\right]$, and in $\mathrm{ℂ}\left[x\right]$.

Short answer

The factorization in $\mathrm{\mathbb{Q}}\left[x\right]$is $x^4-4=\left(x^2-2\right)\left(x^2+2\right)$.

The factorization in$\mathrm{ℝ}\left[x\right]$is role="math" localid="1648647843738" $x^4-4-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)\left(x^2+2\right)$.

The factorization in role="math" localid="1648646692368" $\mathrm{ℂ}\left[x\right]$is $x^4-4-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)\left(x+i\sqrt{2}\right)\left(x-i\sqrt{2}\right)$.

Step-by-step solution

Determine x4-4  

Consider the given function,$x^4-4$

The factorization in $\mathrm{\mathbb{Q}}\left[x\right]$is $x^4-4=\left(x^2-2\right)\left(x^2+2\right)$.

In $\mathrm{ℝ}\left[x\right],x^2-2-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)$, the factorization is,

$x^4-4-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)\left(x^2+2\right)$.

Determine  x4-4in ℂx

Now, in $\mathrm{ℂ}\left[x\right]$, $x^2+2-\left(x+i\sqrt{2}\right)\left(x-i\sqrt{2}\right)$the factorization is,

$x^4-4-\left(x+\sqrt{2}\right)\left(x-\sqrt{2}\right)\left(x+i\sqrt{2}\right)\left(x-i\sqrt{2}\right)$.