Questions: Unique Factorization Domains

Question 1

The ring ℤ[x] of polynomials with integer coefficients is a UFD but not a PID, because the ideal (2, x) is not principal. What does this tell you about the relationship between PIDs and UFDs?

• A. ℤ[x] cannot be a UFD if it is not a PID — the claim must be wrong
• B. UFDs are a strictly larger class than PIDs: every PID is a UFD, but there exist UFDs that are not PIDs
• C. PIDs and UFDs are incomparable properties — neither implies the other
• D. ℤ[x] is a UFD only because x is transcendental; algebraic extensions of ℤ are never UFDs

Answer: (B)

The hierarchy is fields ⊂ Euclidean domains ⊂ PIDs ⊂ UFDs ⊂ integral domains, and each inclusion is strict. Every PID is a UFD (the ideal structure of PIDs forces unique factorization), but the converse fails: ℤ[x] is a UFD (every polynomial over ℤ factors uniquely into irreducibles) but is not a PID because (2, x) requires two generators. This shows UFDs have better factorization structure than typical integral domains, but not necessarily the strong ideal structure of PIDs.

Question 2

In which of the following rings does the element 6 fail to have a unique factorization into irreducibles?

• A. ℤ (the integers)
• B. ℤ[i] (the Gaussian integers)
• C. ℤ[√−5]
• D. ℤ[x] (polynomials with integer coefficients)

Answer: (C)

In ℤ[√−5], we have 6 = 2 × 3 = (1 + √−5)(1 − √−5), and these are genuinely distinct factorizations into irreducibles — the factors are not associates of each other (no unit relates them). Both ℤ and ℤ[i] are Euclidean domains, hence PIDs, hence UFDs, so factorization in them is unique. ℤ[x] is also a UFD. The pathological example ℤ[√−5] is the canonical demonstration that unique factorization is not automatic in algebraic number rings.

Question 3

Every principal ideal domain (PID) is a unique factorization domain.

• T. True
• F. False

Answer: True

This is a fundamental theorem in ring theory. The ideal structure of a PID — specifically the ascending chain condition on ideals and the fact that every prime ideal is maximal — forces the two conditions of a UFD: every nonzero non-unit factors into irreducibles (existence), and the factorization is unique up to order and units. The proof uses the fact that in a PID, irreducible elements and prime elements coincide, which is the key property that guarantees uniqueness.

Question 4

Nearly every UFD is also a PID.

• T. True
• F. False

Answer: False

The inclusion is strict: PIDs ⊂ UFDs, but not every UFD is a PID. The ring ℤ[x] is the standard counterexample — it has unique factorization (every polynomial over ℤ factors uniquely into irreducible polynomials), but the ideal (2, x) = {2f(x) + x·g(x) : f, g ∈ ℤ[x]} is not principal. No single polynomial generates all elements of this ideal. Unique factorization is a property of elements; being a PID is a property of ideals — and the latter is a stronger structural requirement.

Question 5

Why does ℤ[√−5] fail to be a UFD? Identify the two distinct factorizations of 6 and explain why they cannot be considered equivalent.

Model answer: In ℤ[√−5], 6 factors as 2 × 3 and also as (1 + √−5)(1 − √−5), since (1 + √−5)(1 − √−5) = 1 + 5 = 6. These are genuinely distinct because the four elements 2, 3, (1 + √−5), (1 − √−5) are all irreducible (verified by the norm N(a + b√−5) = a² + 5b²: no element has norm 2 or 3) and no two of them are associates (there are no units other than ±1, and none of the four are ±1 multiples of each other). Unique factorization requires factorizations to agree up to order and units, but there is no unit relating the two factorizations.

The failure of unique factorization in ℤ[√−5] stems from the divergence of irreducible and prime elements in that ring. In a UFD, these must coincide. The element 2 is irreducible in ℤ[√−5] (norm 4, no factorization into elements of norm 2) but is NOT prime: 2 divides (1 + √−5)(1 − √−5) = 6, but 2 divides neither factor. This violation of the prime property is what breaks uniqueness. In ℤ or any PID, irreducible implies prime, which is exactly what guarantees the Fundamental Theorem of Arithmetic.