Exercise · ZW-IRREDUCIBLE-ELEMENT-005 · ★☆☆ Basic
In \(\mathbb{Z}[i]\) with norm \(N(a+bi)=a^2+b^2\), which of these elements is irreducible?
Choose one option.
Show answer
- C. \(3\)
Explanation
If \(3=\alpha\beta\), then \(N(\alpha)N(\beta)=9\), and no element has norm 3 since \(a^2+b^2=3\) has no solution; so one factor has norm 1 and is a unit. The others split: \(5=(2+i)(2-i)\), \(13=(3+2i)(3-2i)\), \(2=(1+i)(1-i)\).
Review the concept: Irreducible element
A nonzero nonunit element \(p\) of an integral domain \(D\) with no proper factorization: in every product of two factors equal to \(p\), one of them is a unit.
\[p=ab\ \Rightarrow\ a\ \text{ or }\ b\ \text{is a unit}\]
More practice problems: irreducible element
- ZW-IRREDUCIBLE-ELEMENT-001 Which of these elements is irreducible in ℤ?
- ZW-IRREDUCIBLE-ELEMENT-002 Which polynomial is irreducible in ℚ[x]?
- ZW-IRREDUCIBLE-ELEMENT-003 Which polynomial is reducible in ℚ[x]?
- ZW-IRREDUCIBLE-ELEMENT-004 Which polynomial is irreducible in ℝ[x]?
- ZW-IRREDUCIBLE-ELEMENT-006 In ℤ[√-5], with N(a+b√-5)=a^2+5b^2, which element is irreducible?
- ZW-IRREDUCIBLE-ELEMENT-007 In ℤ[√-5], the element 2 divides 6=(1+√-5)(1-√-5) but divides neither factor. What follows?
- ZW-IRREDUCIBLE-ELEMENT-008 In ℤ, how many elements are associates of 5 (including 5 itself)?
- ZW-IRREDUCIBLE-ELEMENT-009 A polynomial f∈ℚ[x] of degree 3 is irreducible if and only if: