The ring ℤ[√-5] is not a UFD because 6 = 2 × 3 = (1+√-5)(1-√-5) are distinct element factorizations. However, ℤ[√-5] is a Dedekind domain. What does this mean concretely?
AUnique factorization of elements is secretly valid if you allow complex units
BEvery nonzero ideal of ℤ[√-5] factors uniquely as a product of prime ideals, restoring a form of unique factorization at the ideal level
CThe ring is actually a PID when you localize at every prime
Dℤ[√-5] has unique factorization after adjoining finitely many new elements
In ℤ[√-5], the ideal (6) factors as (2, 1+√-5)²(3, 1+√-5)(3, 1-√-5) — a product of prime ideals. The two element factorizations 2·3 and (1+√-5)(1-√-5) correspond to different groupings of these prime ideal factors. This unique factorization of ideals is the defining feature of Dedekind domains and was Dedekind's great insight: when elements don't factor uniquely, pass to ideals, which always do.
Question 2 Multiple Choice
Which of the following is NOT a characterization of Dedekind domains?
ANoetherian, integrally closed, dimension 1
BEvery nonzero ideal factors uniquely into prime ideals
CEvery nonzero fractional ideal is invertible
DEvery ideal is principal
The first three are equivalent characterizations of Dedekind domains. The fourth describes PIDs, which are a strictly smaller class. Every PID is a Dedekind domain (it is Noetherian, integrally closed, and has dimension 1 since nonzero primes are maximal), but Dedekind domains like ℤ[√-5] need not be PIDs. The class group of a Dedekind domain — the group of fractional ideals modulo principal ones — measures exactly how far the domain is from being a PID.
Question 3 True / False
Every PID is a Dedekind domain.
TTrue
FFalse
Answer: True
A PID is an integral domain where every ideal is principal. It is automatically Noetherian (every ideal has one generator), integrally closed (PIDs are UFDs, and UFDs are integrally closed), and one-dimensional (every nonzero prime ideal is maximal in a PID). These three conditions are exactly the definition of a Dedekind domain. The converse fails: ℤ[√-5] is a Dedekind domain but not a PID.
Question 4 True / False
The class group of a Dedekind domain is trivial if and only if the domain is a PID.
TTrue
FFalse
Answer: True
The class group Cl(R) of a Dedekind domain R is the group of fractional ideals modulo principal fractional ideals. It is trivial — consisting of just the identity — precisely when every fractional ideal is principal, which happens if and only if every ideal is principal (i.e., R is a PID). The class number |Cl(R)| measures the failure of unique factorization at the element level. For ℤ[√-5], the class number is 2.
Question 5 Short Answer
Explain why unique factorization of ideals in a Dedekind domain can be viewed as 'rescuing' the unique factorization that fails at the element level.
Think about your answer, then reveal below.
Model answer: In a UFD, every element factors uniquely into irreducibles. In a Dedekind domain that is not a UFD (like ℤ[√-5]), elements can have multiple genuinely distinct factorizations. But every nonzero ideal factors uniquely into prime ideals. The principal ideal (6) in ℤ[√-5] factors into four prime ideals, and the two element factorizations 2·3 and (1+√-5)(1-√-5) correspond to grouping those prime ideal factors differently. The 'lost' uniqueness at the element level is recovered at the ideal level — prime ideals serve as the 'true primes' of the ring.
This perspective was historically revolutionary. Kummer originally introduced 'ideal numbers' to restore unique factorization in cyclotomic rings. Dedekind formalized this as the theory of ideals. The insight is that individual elements may be too coarse to capture the arithmetic structure — ideals, being sets of elements, carry finer information. This is why algebraic number theory is fundamentally about ideals, not elements.