Exercise · ZW-UNIQUE-FACTORIZATION-DOMAIN-009 · ★★★ Advanced
What is the factorization of \(x^4+4\) into irreducibles of \(\mathbb{Q}[x]\)?
Choose one option.
Show answer
- A. \((x^2+2x+2)(x^2-2x+2)\)
Explanation
Write \(x^4+4=(x^2+2)^2-4x^2=(x^2+2x+2)(x^2-2x+2)\), and each quadratic has discriminant \(-4<0\), so it is irreducible. With no rational roots, the degree-4 polynomial is still reducible.
Review the concept: Unique factorization domain
An integral domain \(D\) in which every nonzero nonunit element \(a\) is a product of irreducibles \(p_i\), and this factorization is unique up to order and associates.
\[a=p_1p_2\cdots p_r\]
More practice problems: unique factorization domain
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