Express $x_{4} - 4$ as a product of irreducible in $ℚ [x]$ , in role="math" localid="1648646593814" $ℝ [x]$ , and in $ℂ [x]$ .

Question

Accepted Answer

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

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

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

Short Answer