Exercise · ZW-UNIQUE-FACTORIZATION-DOMAIN-004 · ★☆☆ Basic
Which of these rings is a unique factorization domain?
Choose one option.
Show answer
- C. \(\mathbb{Z}[i]\)
Explanation
The ring \(\mathbb{Z}[i]\) is Euclidean with norm \(a^2+b^2\), hence a UFD. In the others uniqueness fails: \(6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})\), \(4=2\cdot2=(1+\sqrt{-3})(1-\sqrt{-3})\) and \(6=2\cdot3=-(\sqrt{-6})^2\).
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
- ZW-UNIQUE-FACTORIZATION-DOMAIN-001 In ℤ, what is the factorization of 60 into positive irreducibles?
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- ZW-UNIQUE-FACTORIZATION-DOMAIN-003 Using 360=2^3·3^2·5 and 84=2^2·3·7, what is the greatest common divisor of 360 and 84?
- ZW-UNIQUE-FACTORIZATION-DOMAIN-005 Which implication between types of domains is correct?
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- ZW-UNIQUE-FACTORIZATION-DOMAIN-007 Which ideal of ℤ[x] shows that ℤ[x] is not a principal ideal domain, even though it is a UFD?
- ZW-UNIQUE-FACTORIZATION-DOMAIN-008 In ℚ[x], how many irreducible factors (with multiplicity, up to associates) does x^4-x^2 have?
- ZW-UNIQUE-FACTORIZATION-DOMAIN-009 What is the factorization of x^4+4 into irreducibles of ℚ[x]?
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