Exercise · ZW-UNIQUE-FACTORIZATION-DOMAIN-005 · ★☆☆ Basic
Which implication between types of domains is correct?
Choose one option.
Show answer
- A. Every principal ideal domain is a UFD
Explanation
Every principal ideal domain is a UFD, though the chain does not reverse: \(\mathbb{Z}[x]\) is a UFD without being a principal ideal domain, and \(\mathbb{Z}[\sqrt{-5}]\) is an integral domain that is not a UFD.
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?
- ZW-UNIQUE-FACTORIZATION-DOMAIN-002 What is the factorization of x^4-1 into irreducibles of ℚ[x]?
- 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-004 Which of these rings is a unique factorization domain?
- ZW-UNIQUE-FACTORIZATION-DOMAIN-006 In ℤ[√-5], the element 6 has the factorizations 2·3 and (1+√-5)(1-√-5). Since the four factors are…
- 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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