01.02.07 · foundations / groups

Polynomial rings, PIDs, UFDs, and Euclidean domains

shipped3 tiersLean: none

Anchor (Master): Gauss 1801 Disquisitiones Arithmeticae; Kummer 1846 J. reine angew. Math. 35; Dedekind 1871 supplement XI; Hilbert 1890 Math. Ann. 36; Eisenstein 1850 J. reine angew. Math. 39; Motzkin 1949 Bull. AMS 55; Auslander-Buchsbaum 1959 Proc. Natl. Acad. Sci.; Lang Algebra 3e §II.5, V; Atiyah-Macdonald §1, 6, 10; Eisenbud §0-4

Intuition Beginner

The integers have a famous property: every whole number bigger than one breaks into prime factors in exactly one way (apart from rearranging them). The number $12$ is $2 \times 2 \times 3$ , and nothing else. This single-recipe property is what people mean when they say arithmetic with whole numbers is well behaved.

This unit asks which other number-like systems share the same single-recipe property, and which fail it. The answer organises rings into a nested chain of three classes, from strongest to weakest. The strongest class has a division algorithm (a way to compute remainders), the middle class has every ideal generated by a single element, and the weakest class still has unique factorisation into building blocks even when the first two properties break.

A useful image: the integers sit inside the rationals inside the reals inside the complex numbers. Polynomials with real coefficients form a parallel ring next to the integers, and they too have a division algorithm with remainders of smaller degree. The Gaussian integers $a + bi$ with $a, b$ whole join the family. These three rings share a common structural skeleton.

Why does this concept exist? Because uniqueness of factorisation is the silent assumption behind every calculation involving primes, polynomial roots, and ratios, and we need to know exactly which rings actually have it.

Visual Beginner

The picture shows three nested circles labelled UFD, PID, and Euclidean from outside to inside. The integers, polynomials over a field, and Gaussian integers sit inside the smallest circle. Some rings sit between the middle and inner circles, and others between the outer and middle circles.

The nesting captures the structural hierarchy: every Euclidean domain is a principal ideal domain, every principal ideal domain has unique factorisation, but each containment is strict, with explicit examples in every band.

Worked example Beginner

Factor the polynomial $x^{4} - 1$ inside the ring $R [x]$ of polynomials with real coefficients, and then again inside the larger ring $C [x]$ of polynomials with complex coefficients.

Step 1. Recognise $x^{4} - 1 = (x^{2} - 1) (x^{2} + 1)$ using the difference-of-squares identity applied to $x^{2}$ .

Step 2. Apply difference of squares to the first factor: $x^{2} - 1 = (x - 1) (x + 1)$ . The second factor $x^{2} + 1$ has no real roots, so over the real numbers it stays put.

Step 3. Over the real numbers the factorisation is $x^{4} - 1 = (x - 1) (x + 1) (x^{2} + 1)$ . Over the complex numbers the factor $x^{2} + 1$ splits further as $(x - i) (x + i)$ , giving $x^{4} - 1 = (x - 1) (x + 1) (x - i) (x + i)$ .

What this tells us: the same polynomial factors into different pieces depending on which ring we work inside, but each ring gives a single-recipe factorisation into its own irreducible building blocks.

Check your understanding Beginner

Exercise (easy, multiple choice).

Which of the following rings has a division algorithm with a remainder of smaller size?

A. The ring of polynomials in two variables over the real numbers

B. The ring of polynomials in one variable over the real numbers

C. The ring $Z [- 5]$ of numbers of the form $a + b - 5$ with $a, b$ whole

D. The ring of formal power series in two variables

Hint

A division algorithm needs a notion of size that strictly decreases when you take a remainder. For polynomials in one variable, the degree works.

Answer

B. The ring of polynomials in one variable over the real numbers.

Feedback-correct: dividing $f$ by $g$ in $R [x]$ produces a remainder $r$ with degree strictly less than the degree of $g$ . Feedback-wrong: option A has no single-variable degree that works for division; option C lacks unique factorisation altogether; option D has no division-with-remainder of decreasing size.

Formal definition Intermediate+

This unit introduces three nested classes of integral domains. Throughout, a domain means a commutative ring with $1 \neq = 0$ and no zero divisors: if $ab = 0$ then $a = 0$ or $b = 0$ .

Definition (Euclidean domain). A domain $D$ is a Euclidean domain if there exists a function $N : D ∖ {0} \to N$ , called a Euclidean norm or size function, such that for every $a, b \in D$ with $b \neq = 0$ there exist $q, r \in D$ with $a = q b + r$ and either $r = 0$ or $N (r) < N (b)$ ^{[Dummit-Foote §8.1]}.

Definition (principal ideal domain). A domain $D$ is a principal ideal domain (PID) if every ideal of $D$ is principal: for every ideal $I \subseteq D$ there is $a \in D$ with $I = (a) = {r a : r \in D}$ ^{[Lang §II.5]}.

Definition (unique factorisation domain). A domain $D$ is a unique factorisation domain (UFD) if every nonzero non-unit $a \in D$ admits a factorisation $a = u \cdot p_{1} \dots p_{n}$ with $u$ a unit and each $p_{i}$ irreducible, and any two such factorisations agree up to ordering of factors and multiplication of each factor by a unit ^{[Atiyah-Macdonald §1]}.

Recall the divisibility vocabulary inside a domain. An element $u \in D$ is a unit if $uv = 1$ for some $v \in D$ . Elements $a, b$ are associates, written $a \sim b$ , if $a = u b$ for a unit $u$ . A nonzero non-unit $p \in D$ is irreducible if $p = ab$ forces one of $a, b$ to be a unit; it is prime if $p ∣ ab$ forces $p ∣ a$ or $p ∣ b$ . In any domain, prime implies irreducible, but the converse can fail; the converse holds exactly in unique factorisation domains.

The four canonical Euclidean domains are the integers $Z$ with $N (n) = ∣ n ∣$ , the polynomial ring $k [x]$ over a field $k$ with $N (p) = de g p$ , the Gaussian integers $Z [i]$ with $N (a + bi) = a^{2} + b^{2}$ , and the Eisenstein integers $Z [ω]$ with $ω = e^{2 π i /3}$ and $N (a + bω) = a^{2} - ab + b^{2}$ . Each verification follows the same template as the Gaussian-integers case from the predecessor unit's exercises: divide in the ambient field, round each coordinate to the nearest lattice element, and bound the remainder norm by half (or less) of the divisor norm.

The implication chain runs Euclidean domain $\Rightarrow$ PID $\Rightarrow$ UFD, and each containment is strict. The first implication uses the Euclidean algorithm to extract a single generator from any ideal: let $b \in I$ be a nonzero element of minimum norm, and apply Euclidean division to show every element of $I$ is a multiple of $b$ . The second implication uses noetherian induction on the ascending chain of principal ideals combined with the equivalence prime $\Leftrightarrow$ irreducible in a PID, which is the foundational step bridging existence to uniqueness of factorisation.

Counterexamples to common slips Intermediate+

Irreducible does not imply prime in a general domain. Inside $Z [- 5]$ the element $2$ is irreducible (no factorisation into non-units exists) but not prime: $2 ∣ (1 + - 5) (1 - - 5) = 6$ without $2$ dividing either factor. The unique-factorisation failure $6 = 2 \cdot 3 = (1 + - 5) (1 - - 5)$ witnesses both the prime-versus-irreducible gap and the lack of UFD structure.
A UFD need not be a PID. The ring $Z [x]$ is a UFD by Gauss's lemma but not a PID: the ideal $(2, x)$ requires two generators. Geometrically, the closed point corresponding to the maximal ideal $(2, x)$ has codimension $2$ in $Spec Z [x]$ , and codimension-two closed points cannot be cut out by a single equation.
A PID need not be Euclidean. The ring $Z [(1 + - 19) /2]$ is a PID (verified via the Minkowski bound on its class number) but admits no Euclidean norm, by Motzkin's 1949 explicit construction.

Key theorem with proof Intermediate+

Theorem (Euclidean $\Rightarrow$ PID $\Rightarrow$ UFD). Let $D$ be a Euclidean domain. Then $D$ is a principal ideal domain. Let $D$ be a principal ideal domain. Then $D$ is a unique factorisation domain.

Proof. Part 1: Euclidean $\Rightarrow$ PID. Let $D$ be a Euclidean domain with norm $N$ and let $I \subseteq D$ be a nonzero ideal. The set ${N (a) : a \in I, a \neq = 0}$ is a nonempty subset of $N$ and hence has a least element. Choose $b \in I$ with $b \neq = 0$ and $N (b)$ minimal. We claim $I = (b)$ . The inclusion $(b) \subseteq I$ holds because $b \in I$ . For the reverse, let $a \in I$ . Apply the Euclidean division: $a = q b + r$ with $r = 0$ or $N (r) < N (b)$ . Then $r = a - q b \in I$ . If $r \neq = 0$ then $N (r) < N (b)$ , contradicting the minimal choice of $b$ . Hence $r = 0$ and $a = q b \in (b)$ . Therefore $I = (b)$ is principal.

Part 2: PID $\Rightarrow$ UFD. Let $D$ be a PID. We establish two intermediate results: (a) every nonzero non-unit factors as a finite product of irreducibles; (b) in a PID, every irreducible is prime. Uniqueness then follows from (b).

For (a), suppose some nonzero non-unit $a_{0}$ admits no factorisation into irreducibles. Then $a_{0}$ is reducible, say $a_{0} = a_{1} b_{1}$ with both $a_{1}, b_{1}$ non-units. At least one of $a_{1}, b_{1}$ admits no factorisation into irreducibles; call it $a_{1}$ . Iterate to produce $a_{0}, a_{1}, a_{2}, \dots$ with $a_{n + 1}$ a proper divisor of $a_{n}$ , hence $(a_{n}) ⊊ (a_{n + 1})$ . The union $⋃_{n} (a_{n})$ is an ideal, hence principal by the PID hypothesis, say $⋃_{n} (a_{n}) = (c)$ . Then $c \in (a_{N})$ for some $N$ , so $(c) \subseteq (a_{N}) \subseteq (c)$ , forcing the chain to stabilise at $N$ , contradicting strict inclusion. Hence every nonzero non-unit factors into irreducibles.

For (b), let $p \in D$ be irreducible and suppose $p ∣ ab$ with $p ∤ a$ . The ideal $(p, a)$ is principal, say $(p, a) = (d)$ . Then $d ∣ p$ , so $d$ is either a unit or an associate of $p$ . If $d \sim p$ then $p ∣ a$ , contrary to assumption. Hence $d$ is a unit, so $(p, a) = D$ , which gives $1 = r p + s a$ for some $r, s \in D$ . Multiplying by $b$ : $b = r p b + s ab$ . Since $p ∣ ab$ , both terms on the right are divisible by $p$ , so $p ∣ b$ . Therefore $p$ is prime.

To conclude uniqueness, suppose $u p_{1} \dots p_{m} = v q_{1} \dots q_{n}$ with $u, v$ units and each $p_{i}, q_{j}$ irreducible. The element $p_{1}$ is prime by (b), so $p_{1} ∣ q_{j}$ for some $j$ . Since $q_{j}$ is irreducible and $p_{1}$ is non-unit, $p_{1} \sim q_{j}$ . Cancel and induct on $m + n$ . This identifies the two factorisations up to ordering and units. $□$

Bridge. The chain Euclidean $\Rightarrow$ PID $\Rightarrow$ UFD builds toward 21.02.03 $p$ -adic numbers, where the discrete-valuation-ring structure of $Z_{p}$ exhibits the most rigid Euclidean shape of all, and appears again in 04.02.07 Nullstellensatz, where the UFD structure of $k [x_{1}, \dots, x_{n}]$ underwrites the radical-ideal correspondence. The foundational reason is that the Euclidean division algorithm generates principal ideals by minimality, and this is exactly the structure that promotes irreducible elements to prime elements. The bridge is between the existence half (no infinite descending chain of proper divisors, equivalent to noetherianity on principal ideals) and the uniqueness half (prime equals irreducible). Putting these together with Gauss's lemma at Master tier, the construction generalises from one-variable polynomial rings to many-variable polynomial rings, so $k [x_{1}, \dots, x_{n}]$ inherits unique factorisation by iterating the lift $D \mapsto D [x]$ .

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Show that the ideal $(x, y) \subset k [x, y]$ , where $k$ is a field, is not principal.

Hint

Suppose for contradiction that $(x, y) = (f)$ for some polynomial $f \in k [x, y]$ . Show $f$ must divide both $x$ and $y$ , then constrain the degree of $f$ .

Answer

Assume $(x, y) = (f)$ . Then $f ∣ x$ and $f ∣ y$ in $k [x, y]$ . From $f ∣ x$ , viewing both as polynomials in $x$ with coefficients in $k [y]$ , the divisor $f$ has $x$ -degree at most $1$ . Symmetrically, $f$ has $y$ -degree at most $1$ , with mixed term ruled out by total degree considerations. Combined with $f ∣ x$ , the only candidate is $f = c$ for a nonzero $c \in k$ , but then $f$ is a unit and $(f) = k [x, y]$ , which contains $1$ . However, $1 \in / (x, y)$ : every element of $(x, y)$ has zero constant term, while $1$ does not. The contradiction shows $(x, y)$ is not principal, so $k [x, y]$ is not a PID. It remains a UFD by Gauss's lemma.

Exercise 5 (medium, symbolic).

Show that $Z [- 5]$ is not a unique factorisation domain by exhibiting two essentially different factorisations of $6$ .

Hint

Use the norm $N (a + b - 5) = a^{2} + 5 b^{2}$ . Show that $2$ , $3$ , $1 + - 5$ , and $1 - - 5$ are all irreducible by computing their norms.

Answer

Two factorisations are $6 = 2 \cdot 3$ and $6 = (1 + - 5) (1 - - 5)$ . The norm $N (a + b - 5) = a^{2} + 5 b^{2}$ is multiplicative. The norms of the four factors are $N (2) = 4$ , $N (3) = 9$ , $N (1 + - 5) = 6$ , $N (1 - - 5) = 6$ .

The only norms attainable by non-units in $Z [- 5]$ are $4, 5, 6, 9, \dots$ (those of the form $a^{2} + 5 b^{2}$ excluding $1$ ). There is no element of norm $2$ or $3$ . So if $2 = α β$ with $α, β$ non-units, then $N (α) N (β) = 4$ with each norm at least $2$ , forcing $N (α) = N (β) = 2$ , impossible. Hence $2$ is irreducible. Similarly $3$ is irreducible (no element of norm $3$ ). For $1 \pm - 5$ with norm $6$ , a non-degenerate factorisation would need norms multiplying to $6$ , so ${2, 3}$ , again impossible. All four factors are irreducible. The two factorisations cannot be matched up by units (the units of $Z [- 5]$ are just $\pm 1$ , since $u \overset{u}{ˉ} = N (u) = 1$ forces $u = \pm 1$ ), so unique factorisation fails.

Exercise 7 (hard, symbolic).

Let $D$ be a PID and $p \in D$ a nonzero non-unit. Prove that the ideal $(p)$ is maximal in $D$ if and only if $p$ is irreducible.

Hint

For the forward direction, suppose $(p)$ is maximal and $p = ab$ . Show $(b) = D$ or $(b) = (p)$ . For the reverse, suppose $p$ is irreducible. Show that any ideal $I$ with $(p) \subseteq I \subseteq D$ equals either $(p)$ or $D$ .

Answer

( $\Leftarrow$ ) Suppose $p$ is irreducible. Let $I$ be an ideal with $(p) \subseteq I \subseteq D$ . Since $D$ is a PID, $I = (b)$ for some $b \in D$ . From $(p) \subseteq (b)$ we get $b ∣ p$ . By irreducibility of $p$ , either $b$ is a unit (so $I = (b) = D$ ) or $b$ is an associate of $p$ (so $I = (b) = (p)$ ). Hence no ideal lies strictly between $(p)$ and $D$ , and $(p)$ is maximal.

( $\Rightarrow$ ) Suppose $(p)$ is maximal. Write $p = ab$ with $a, b \in D$ . Then $(p) \subseteq (a)$ . By maximality, $(a) = (p)$ or $(a) = D$ . If $(a) = (p)$ then $a$ and $p$ are associates, so $b$ is a unit. If $(a) = D$ then $a$ is a unit. Either way, one of $a, b$ is a unit, so $p$ is irreducible.

Exercise 8 (hard, symbolic).

Prove that in a UFD, every irreducible element is prime.

Hint

Let $p$ be irreducible and suppose $p ∣ ab$ . Write the unique factorisations of $a$ and $b$ , multiply them, and compare with the unique factorisation of $ab$ , using $p$ as a factor.

Answer

Let $D$ be a UFD and $p \in D$ irreducible. Suppose $p ∣ ab$ , say $ab = p c$ for some $c \in D$ . We show $p ∣ a$ or $p ∣ b$ .

If $a = 0$ or $b = 0$ , the conclusion is direct. Otherwise, write the unique factorisations $a = u_{a} \prod_{i} q_{i}$ , $b = u_{b} \prod_{j} r_{j}$ , and $c = u_{c} \prod_{k} s_{k}$ , where $u_{a}, u_{b}, u_{c}$ are units and $q_{i}, r_{j}, s_{k}$ are irreducibles. Then $ab = u_{a} u_{b} \prod_{i} q_{i} \prod_{j} r_{j}$ and also $p c = u_{c} p \prod_{k} s_{k}$ . By unique factorisation applied to $ab = p c$ , the multiset of irreducible factors (up to associates) of $ab$ equals that of $p c$ . Hence $p$ is an associate of some $q_{i}$ or some $r_{j}$ . In the first case $p ∣ a$ ; in the second $p ∣ b$ . Therefore $p$ is prime.

In a general domain, prime always implies irreducible, but the converse uses the uniqueness hypothesis: without it (as in $Z [- 5]$ ), the converse fails.

Advanced results Master

Theorem 1 (Gauss's lemma). If $D$ is a UFD then the polynomial ring $D [x]$ is a UFD. By iteration, $D [x_{1}, \dots, x_{n}]$ is a UFD whenever $D$ is. In particular, $Z [x_{1}, \dots, x_{n}]$ and $k [x_{1}, \dots, x_{n}]$ for $k$ a field are UFDs ^{[Gauss 1801, art. 42]}. The classical statement is that the product of two primitive polynomials in $Z [x]$ is primitive, and Kronecker 1881 extended this to general UFDs. The proof rests on the content-primitive decomposition: every $f \in D [x]$ factors as $f = c (f) \cdot f^{*}$ where $c (f) \in D$ is the content (gcd of coefficients) and $f^{*}$ is primitive. Multiplicativity of content under multiplication of primitives, combined with the UFD structure of $D$ for contents and of $Frac (D) [x]$ for primitives, lifts unique factorisation to $D [x]$ .

Gauss's lemma is the bridge between number-theoretic UFD structure and polynomial UFD structure: it explains why $Z [x_{1}, \dots, x_{n}]$ shares the unique-factorisation property of $Z$ even when it loses the principal-ideal property. The polynomial-ring closure operation $D \mapsto D [x]$ preserves UFD but does not preserve PID in general: $Z [x]$ is a UFD but the ideal $(2, x)$ is not principal. The closure operation does preserve PID when restricted to fields: $k [x]$ for $k$ a field is a Euclidean domain, hence a PID, by the degree algorithm. The contrast manifests algebraically as the failure of $Frac (D [x]) = D (x_{1})$ to coincide with $D$ except in the field case. Gauss's original 1801 statement in Disquisitiones Arithmeticae art. 42 was about $Z [x]$ ; Kronecker generalised to UFDs and the modern formulation appears in van der Waerden 1930.

Theorem 2 (Eisenstein's criterion). Let $D$ be a UFD with field of fractions $K$ , and let $f (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0} \in D [x]$ . If there exists a prime $p \in D$ such that $p ∤ a_{n}$ , $p ∣ a_{i}$ for $0 \leq i < n$ , and $p^{2} ∤ a_{0}$ , then $f$ is irreducible in $K [x]$ . Eisenstein 1850 J. reine angew. Math. 39, 160 ^{[Eisenstein 1850]} introduced this criterion to establish irreducibility of the cyclotomic polynomial $Φ_{p} (x) = (x^{p} - 1) / (x - 1)$ for prime $p$ , after the substitution $x \mapsto x + 1$ produces a polynomial satisfying the criterion. Schönemann 1846 J. reine angew. Math. 32, 93 stated a closely related criterion two years earlier; the dual attribution is sometimes written as the Schönemann-Eisenstein criterion.

The proof reduces modulo $p$ : in the quotient $D / p \cdot D [x]$ , the polynomial $\overset{ˉ}{f}$ becomes $\overset{a}{ˉ}_{n} x^{n}$ . If $f = g h$ in $D [x]$ with $g, h$ non-constant, then $\overset{g}{ˉ} \overset{ˉ}{h} = \overset{a}{ˉ}_{n} x^{n}$ in the integral domain $(D / p) [x]$ , forcing both $\overset{g}{ˉ}$ and $\overset{ˉ}{h}$ to be monomials in $x$ . Their constant terms vanish modulo $p$ , so the constant term of $f = g h$ is divisible by $p^{2}$ , contradicting $p^{2} ∤ a_{0}$ . Generalisations include Eisenstein at prime ideals of Dedekind domains (Newton polygon criterion), and the Krasner lemma in $p$ -adic analysis. Modern applications include rapid irreducibility checks in computer algebra systems (Magma, Pari/GP, SageMath) and the cyclotomic-polynomial irreducibility proof that opens Galois theory of cyclotomic fields.

Theorem 3 (Strict hierarchy: ED $⊊$ PID $⊊$ UFD). Each containment in the chain Euclidean domain $\subseteq$ PID $\subseteq$ UFD is strict. The strict separation PID $⊊$ UFD is witnessed by $Z [x]$ , which is a UFD by Gauss's lemma but not a PID (the ideal $(2, x)$ requires two generators). The strict separation ED $⊊$ PID is witnessed by $Z [α]$ with $α = (1 + - 19) /2$ : this ring is a PID (its class number equals $1$ by the Minkowski-bound argument applied to its ring of integers $O_{K}$ inside $K = Q (- 19)$ ) but admits no Euclidean norm. Motzkin 1949 Bull. AMS 55, 1142 ^{[Motzkin 1949]} constructed the explicit obstruction: every Euclidean domain must contain a so-called universal side divisor (a non-unit $b$ such that every element is congruent modulo $b$ to either $0$ or a unit); $Z [(1 + - 19) /2]$ has no such element, so it is not Euclidean despite being a PID.

The Motzkin obstruction is the most subtle of the three hierarchy separations. The standard proof proceeds by examining elements of small norm: in $Z [(1 + - 19) /2]$ , the only units are $\pm 1$ , and the elements of next-smallest norm are $\pm 2$ and $\pm 3$ and a few others. A universal side divisor among these candidates would have to make every residue class either $0$ or $\pm 1$ modulo it, and a finite case-check rules each candidate out. Lenstra 1979 Acta Arith. 35 surveyed the broader question of when imaginary quadratic rings of integers are Euclidean for any norm: only nine values of $d < 0$ make $Q (d)$ 's ring of integers Euclidean, namely $d \in {- 1, - 2, - 3, - 7, - 11}$ for the norm function and $d \in {- 1, - 2, - 3, - 7, - 11, - 19, - 43, - 67, - 163}$ for PID-ness (the famous Stark-Heegner list of nine imaginary quadratic fields with class number one).

Theorem 4 (Class group and ideal-theoretic measure of UFD failure). Let $O_{K}$ be the ring of integers of an algebraic number field $K$ . The ideal class group $Cl (O_{K})$ is the quotient of the group of fractional ideals by the subgroup of principal fractional ideals. Then $Cl (O_{K}) = {e}$ if and only if $O_{K}$ is a PID, equivalently a UFD (since Dedekind domains are PIDs precisely when they are UFDs). The class group is finite, with size $h_{K}$ bounded by the Minkowski bound $M_{K} = ∣ d_{K} ∣ (4/ π)^{r_{2}} n! / n^{n}$ , where $d_{K}$ is the discriminant, $n = [K : Q]$ , and $r_{2}$ the number of complex places. Class-group computations give: $Cl (Z [- 5]) ≅ Z /2 Z$ generated by the non-principal prime ideal $(2, 1 + - 5)$ , whose square is principal: $(2, 1 + - 5)^{2} = (2)$ . This is exactly the algebraic reason for the non-unique factorisation $6 = 2 \cdot 3 = (1 + - 5) (1 - - 5)$ : at the level of ideals the factorisation becomes unique, $(6) = p_{2}^{2} p_{3} \overset{ˉ}{p}_{3}$ , with $p_{2} = (2, 1 + - 5)$ and $p_{3}, \overset{ˉ}{p}_{3}$ the two primes above $3$ .

Kummer 1846 J. reine angew. Math. 35, 327 ^{[Kummer 1846]} discovered the unique-factorisation failure in cyclotomic rings $Z [ζ_{p}]$ for $p = 23$ and beyond while attempting Fermat's Last Theorem, and introduced "ideal numbers" to restore unique factorisation. Dedekind 1871 supplement XI to Dirichlet's Vorlesungen ^{[Dedekind 1871]} reformulated Kummer's ideal numbers as the modern ideal-theoretic concept, recovering unique factorisation at the level of ideals: every nonzero ideal in $O_{K}$ factors uniquely as a product of prime ideals (the Dedekind unique-factorisation theorem). The class group records exactly how far the ring is from being a PID, and the finiteness of $Cl (O_{K})$ is the substantive content of Minkowski's geometry-of-numbers argument 1896 Geometrie der Zahlen. Iwasawa 1959-1973 organised the variation of class groups along $Z_{p}$ -extensions, opening the modern theory of $p$ -adic L-functions and the Iwasawa main conjecture (proved by Mazur-Wiles 1984 for cyclotomic, by Wiles 1990 for totally real).

Theorem 5 (Hilbert basis theorem and Noetherian closure under polynomial extension). Let $R$ be a Noetherian commutative ring. Then the polynomial ring $R [x]$ is Noetherian. By induction, $R [x_{1}, \dots, x_{n}]$ is Noetherian. In particular, $k [x_{1}, \dots, x_{n}]$ for $k$ a field is Noetherian. Hilbert 1890 Math. Ann. 36, 473 ^{[Hilbert 1890]} proved this and used it to settle Gordan's invariant-theory problem, establishing finite generation of the invariant ring of any classical group acting on a finite-dimensional vector space. The proof uses leading coefficients: given an ideal $I \subseteq R [x]$ , the set of leading coefficients of polynomials in $I$ forms an ideal $J \subseteq R$ . Choose finitely many generators of $J$ , lift them to polynomials in $I$ , and combine with the finitely many ideals $I \cap R [x]_{\leq d}$ for $d$ up to the maximum degree of the lifts.

The Hilbert basis theorem propagates noetherian-ness to all rings of practical interest in algebraic geometry. Combined with the UFD-extension Gauss's lemma, it makes $k [x_{1}, \dots, x_{n}]$ simultaneously noetherian and UFD, the two structural conditions undergirding classical algebraic geometry. Buchberger 1965 introduced Gröbner bases as algorithmic finite generating sets, making the Hilbert basis theorem effective in computer algebra. The combinatorial counterpart is the Robbiano-Schwartz theory of monomial orders, where every monomial order on $k [x_{1}, \dots, x_{n}]$ is induced by a real-weight vector composed with a tie-breaking rule.

Theorem 6 (Auslander-Buchsbaum: regular local rings are UFDs). Every regular local ring is a UFD. Auslander-Buchsbaum 1959 Proc. Natl. Acad. Sci. USA 45, 733 ^{[Auslander-Buchsbaum 1959]} settled a long-standing conjecture of Krull by combining the homological characterisation of regularity (Serre 1955: a noetherian local ring is regular if and only if every finitely generated module has finite projective dimension) with the Auslander-Buchsbaum formula relating depth and projective dimension. The proof passes through an induction on the dimension of the regular local ring and the structure theorem for finitely generated modules over a PID applied to the maximal-ideal quotient. Modern proofs use the Koszul complex and Serre's intersection-multiplicity formula.

The Auslander-Buchsbaum theorem is the homological-algebra payoff for the structural distinction between UFDs and merely noetherian domains. The class group of a regular local ring vanishes, while the class group of a non-regular noetherian local domain can be nonzero (e.g., singular toric varieties). Geometrically, regularity at a point means the local ring is the ring of formal power series in $dim$ -many variables over the residue field (by Cohen 1946); this formal-disc structure inherits unique factorisation from the polynomial ring via the standard density argument. The theorem fails outside the noetherian case: there exist non-noetherian regular rings (e.g., the absolute integral closure $\overline{Z}$ in algebraic geometry) that are not UFDs.

Theorem 7 (Hilbert's Nullstellensatz, algebraic form). Let $k$ be an algebraically closed field. Then the maximal ideals of $k [x_{1}, \dots, x_{n}]$ are exactly the ideals $(x_{1} - a_{1}, \dots, x_{n} - a_{n})$ for $(a_{1}, \dots, a_{n}) \in k^{n}$ . The radical correspondence $I (V (J)) = J$ identifies the lattice of radical ideals with the lattice of Zariski-closed subsets of $k^{n}$ . Hilbert 1893 Math. Ann. 42, 313 proved this as the foundational algebra-geometry dictionary, identifying algebraic varieties over $k$ with quotient rings $k [x_{1}, \dots, x_{n}] / I$ for radical ideals $I$ .

The Nullstellensatz is exactly the content of the maximal-ideal structure of $k [x_{1}, \dots, x_{n}]$ for algebraically closed $k$ . The algebraic closure hypothesis is essential: over $R [x]$ , the maximal ideals include $(x^{2} + 1)$ (since $R [x] / (x^{2} + 1) ≅ C$ is a field), not corresponding to any real point. The unit's UFD structure of $k [x_{1}, \dots, x_{n}]$ underwrites the radical correspondence: the radical of an ideal $J$ in a UFD equals the product of distinct prime factors of any generator (when principal), generalising to the union of associated primes in the noetherian case. Zariski-Samuel Commutative Algebra II Ch. VII gives the standard textbook proof via the Noether normalisation lemma; Atiyah-Macdonald ex. 7.14 gives a slicker proof via the Rabinowitsch trick on the polynomial ring $k [x_{1}, \dots, x_{n}, t] / (1 - t f)$ .

Theorem 8 (Cohen-Seidenberg going-up and going-down). Let $R \subseteq S$ be commutative rings with $S$ integral over $R$ . The going-up theorem: every prime $p \subset R$ extends to a prime $q \subset S$ with $q \cap R = p$ , and every chain of primes in $R$ lifts to a chain in $S$ . The going-down theorem (under the additional hypothesis that $R$ is normal): every chain of primes in $R$ descending from $p$ lifts to a descending chain in $S$ containing $q$ . Cohen-Seidenberg 1946 Bull. AMS 52, 252 established these as the bridge between dimension theory and integral extensions. The pair underwrites Krull dimension's invariance under integral extensions: $dim R = dim S$ whenever $S$ is integral over $R$ , and the algebraic-geometry consequence is that finite morphisms preserve dimension.

The going-up and going-down theorems are the structural counterpart to the Nullstellensatz in the non-algebraically-closed regime. They identify integral extensions $R \subseteq S$ with finite surjective morphisms of affine schemes $Spec (S) \to Spec (R)$ , and the lifting properties of prime chains translate into surjectivity of the map on prime spectra. The normal-domain hypothesis for going-down is essential: without it, the descending lift can fail at non-normal points. Geometrically this corresponds to the fact that non-normal varieties can have "false" branches that disappear under normalisation. The normalisation $R$ of a domain $R$ inside $Frac (R)$ is the canonical extension restoring normal behaviour, with the geometric normalisation of an affine variety $Spec (R)$ being $Spec (R) \to Spec (R)$ .

Synthesis. The hierarchy Euclidean $⊊$ PID $⊊$ UFD $⊊$ Noetherian domain $⊊$ integral domain is the foundational reason that commutative algebra organises rings by the failure modes of unique factorisation. The central insight is that each containment-failure is geometric: the failure of UFD shows up as a non-zero class group, the failure of PID shows up as ideals requiring multiple generators (codimension-greater-than-one closed subschemes), and the failure of Euclidean shows up as the absence of an effective division algorithm even when ideals are still principal.

Putting these together with Gauss's lemma and the Hilbert basis theorem, the polynomial-ring closure operation $D \mapsto D [x_{1}, \dots, x_{n}]$ propagates noetherian-ness and UFD-ness from $D$ to all polynomial extensions, while losing the PID property except in the one-variable case over a field. This is exactly the bridge between affine algebraic geometry and commutative algebra: the coordinate rings $k [x_{1}, \dots, x_{n}] / I$ of affine varieties are noetherian UFDs locally at smooth points (by Auslander-Buchsbaum), and the failure of UFD at a singular point identifies the local class group with the divisor class group of the singularity. The pattern generalises in two parallel directions. First, the Dedekind extension restores unique factorisation at the level of ideals when it fails at the level of elements, identifying $Cl (O_{K})$ with the obstruction to PID-ness; this is the foundational structure of algebraic number theory and the Iwasawa main conjecture. Second, the regular-local-ring framework identifies smoothness with finite-projective-dimension homological purity, building toward Cohen-Macaulay and Gorenstein rings in modern algebraic geometry, and connecting to the derived-category and motivic frameworks where the polynomial-ring closure is replaced by simplicial commutative rings or $E_{\infty}$ -ring spectra.

Full proof set Master

Proposition 1 (Gauss's lemma — content multiplicativity). Let $D$ be a UFD with field of fractions $K$ . For $f \in D [x]$ , define the content $c (f)$ as the gcd of the coefficients of $f$ , well-defined up to units. Call $f$ primitive if $c (f) = 1$ . Then for any $f, g \in D [x]$ , $c (f g) = c (f) c (g)$ up to units.

Proof. Reduce to the primitive case. Write $f = c (f) f^{*}$ and $g = c (g) g^{*}$ with $f^{*}, g^{*}$ primitive. Then $f g = c (f) c (g) (f^{*} g^{*})$ , so it suffices to show $f^{*} g^{*}$ is primitive, i.e., that the product of two primitive polynomials is primitive.

Suppose for contradiction that $f^{*} g^{*}$ has content divisible by some prime $p \in D$ . Pass to the quotient $(D / p) [x]$ , which is an integral domain (since $D / p$ is a domain when $p$ is prime in the UFD $D$ , hence prime in $D$ as an element). The image $\overline{f^{*} g^{*}} = \overline{f^{*}} \cdot \overline{g^{*}}$ equals zero in $(D / p) [x]$ , so one of $\overline{f^{*}}, \overline{g^{*}}$ is zero. Then all coefficients of either $f^{*}$ or $g^{*}$ are divisible by $p$ , contradicting primitivity. Hence $f^{*} g^{*}$ is primitive and $c (f g) = c (f) c (g)$ up to units. $□$

Proposition 2 (Gauss's lemma — UFD lifts to polynomial ring). Let $D$ be a UFD. Then $D [x]$ is a UFD.

Proof. Let $K = Frac (D)$ . The ring $K [x]$ is a Euclidean domain (with norm degree), hence a UFD. Given a nonzero non-unit $f \in D [x]$ , factor in $K [x]$ : $f = u q_{1} \dots q_{m}$ with $u \in K$ a unit and $q_{i} \in K [x]$ irreducible. Clear denominators of each $q_{i}$ to write $q_{i} = (a_{i} / b_{i}) q_{i}^{*}$ with $q_{i}^{*} \in D [x]$ primitive and $a_{i}, b_{i} \in D$ . Then $f = u \prod (a_{i} / b_{i}) \prod q_{i}^{*}$ . By content multiplicativity, the product $\prod q_{i}^{*}$ is primitive, so the rational factor $u \prod (a_{i} / b_{i})$ must equal $c (f)$ up to a unit, and lies in $D$ . Factor $c (f) \in D$ into irreducibles (using UFD structure of $D$ ).

This produces a factorisation of $f$ in $D [x]$ as a product of irreducibles: irreducible constants from $D$ , plus primitive polynomial factors that remain irreducible in $D [x]$ because their image in $K [x]$ is irreducible.

For uniqueness, suppose $f = \prod p_{i} \prod q_{j}^{*} = \prod p_{i}^{'} \prod (q_{j}^{*})^{'}$ , with $p_{i}, p_{i}^{'}$ irreducible elements of $D$ and $q_{j}^{*}, (q_{j}^{*})^{'}$ primitive polynomials irreducible in $K [x]$ . Comparing contents gives $\prod p_{i} = u \prod p_{i}^{'}$ for a unit $u$ , so by UFD-ness of $D$ these match up to units. Comparing in $K [x]$ matches the $q_{j}^{*}$ to the $(q_{j}^{*})^{'}$ up to units in $K$ , which restrict to units in $D$ on the primitive parts. Hence the factorisations agree, and $D [x]$ is a UFD. $□$

Proposition 3 (Euclidean algorithm produces Bézout coefficients). Let $D$ be a Euclidean domain with norm $N$ , and let $a, b \in D$ with $b \neq = 0$ . There exists $d = g cd (a, b) \in D$ together with $r, s \in D$ such that $d = r a + s b$ .

Proof. Apply the Euclidean division iteratively: $a = q_{1} b + r_{1}$ with $N (r_{1}) < N (b)$ or $r_{1} = 0$ ; if $r_{1} \neq = 0$ , $b = q_{2} r_{1} + r_{2}$ with $N (r_{2}) < N (r_{1})$ or $r_{2} = 0$ ; continue. The sequence of norms $N (r_{1}) > N (r_{2}) > \dots$ strictly decreases in $N$ , so it terminates at some $r_{n} = 0$ . Then $r_{n - 1}$ divides $r_{n - 2}$ (since $r_{n - 2} = q_{n} r_{n - 1}$ ), hence by descent $r_{n - 1}$ divides each preceding remainder and divides $a, b$ . Conversely, any common divisor of $a, b$ divides each $r_{k}$ by induction, so $r_{n - 1}$ is a greatest common divisor.

To produce the Bézout coefficients, work backwards: $r_{n - 1} = r_{n - 3} - q_{n - 1} r_{n - 2}$ , substitute $r_{n - 2} = r_{n - 4} - q_{n - 2} r_{n - 3}$ , and continue until $r_{n - 1}$ is expressed as an integer combination of $a$ and $b$ . The result is the extended Euclidean algorithm, producing $r, s$ explicitly. $□$

Connections Master

Ring, ring homomorphism, and ideal 01.02.06. This unit supplies the ring framework: rings, ideals, and quotients. The polynomial-ring closure operation $D \mapsto D [x]$ explored at this tier extends the ideal-and-quotient machinery of the predecessor unit, and the first isomorphism theorem proved there underwrites the content-primitive decomposition in Gauss's lemma.
Primes and fundamental theorem of arithmetic 21.01.02. The integers $Z$ are the prototype UFD, and the fundamental theorem of arithmetic is exactly the statement that $Z$ is a UFD. The hierarchy framework of this unit generalises the FTA from $Z$ to all UFDs, with the strict separation PID $⊊$ UFD witnessed by polynomial rings $Z [x]$ , and the strict separation ED $⊊$ PID witnessed by $Z [(1 + - 19) /2]$ .
Unique factorisation of ideals 21.02.03. Builds toward Dedekind's restoration of unique factorisation at the ideal level when it fails at the element level. The class group $Cl (O_{K})$ records exactly the obstruction to PID-ness, and the Dedekind unique-factorisation theorem (every nonzero ideal factors uniquely into prime ideals) generalises the Euclidean and PID structure of this unit to all Dedekind domains.
Hilbert Nullstellensatz 01.02.24 pending. The geometric content of the maximal-ideal structure of $k [x_{1}, \dots, x_{n}]$ explored at Theorem 7 above. The Nullstellensatz identifies the maximal ideals of $k [x_{1}, \dots, x_{n}]$ for algebraically closed $k$ with the points of $k^{n}$ , and the radical correspondence $I (V (J)) = J$ identifies radical ideals with Zariski-closed subsets.
Nullstellensatz 04.02.07. Appears again in the geometric setting of schemes and dimension theory. The Spec-construction sends commutative rings to topological spaces with structure sheaves, and the UFD and Noetherian properties developed here underwrite the local smoothness criteria of affine schemes through the regular-local-ring criterion of Auslander-Buchsbaum.

Historical & philosophical context Master

Carl Friedrich Gauss 1801 Disquisitiones Arithmeticae ^{[Gauss 1801]} gave the first rigorous proof that the integers $Z$ have unique factorisation into primes, establishing what later became the fundamental theorem of arithmetic. Gauss also proved the lemma now bearing his name about the product of primitive polynomials in $Z [x]$ (art. 42) and used it to establish the irreducibility of cyclotomic polynomials. Ernst Kummer 1846 J. reine angew. Math. 35, 327 ^{[Kummer 1846]} discovered that unique factorisation fails in the cyclotomic ring $Z [ζ_{23}]$ while attempting Fermat's Last Theorem, and introduced "ideal numbers" as fictitious factors restoring unique factorisation. Gotthold Eisenstein 1850 J. reine angew. Math. 39, 160 ^{[Eisenstein 1850]} gave the irreducibility criterion now bearing his name, with Theodor Schönemann 1846 having stated a closely related criterion two years earlier.

Richard Dedekind 1871 supplement XI to the fourth edition of Dirichlet's Vorlesungen über Zahlentheorie ^{[Dedekind 1871]} reformulated Kummer's ideal numbers as the modern ideal-theoretic concept and proved the Dedekind unique-factorisation theorem at the ideal level. David Hilbert 1890 Math. Ann. 36, 473 ^{[Hilbert 1890]} proved the basis theorem and used it to settle Gordan's invariant-theory problem; Hilbert 1893 Math. Ann. 42, 313 proved the Nullstellensatz, opening the algebra-geometry dictionary. Emmy Noether 1921 Math. Ann. 83, 24 ^{[Noether 1921]} developed the abstract ideal theory and identified the ascending chain condition as the unifying structural property. Wolfgang Krull 1928 S.-B. Heidelberg. Akad. Wiss. 7, 11 ^{[Krull 1928]} established the Hauptidealsatz and dimension theory.

The hierarchy Euclidean $⊊$ PID $⊊$ UFD was sharpened by Theodore Motzkin 1949 Bull. AMS 55, 1142 ^{[Motzkin 1949]}, who proved that $Z [(1 + - 19) /2]$ is a PID but not Euclidean, the most subtle of the three separations. Maurice Auslander and David Buchsbaum 1959 Proc. Natl. Acad. Sci. USA 45, 733 ^{[Auslander-Buchsbaum 1959]} proved that regular local rings are UFDs, closing a major Krull-era conjecture and identifying smoothness with finite homological dimension. Oscar Zariski and Pierre Samuel Commutative Algebra (1958-1960) gave the canonical synthesis of pre-Grothendieck commutative algebra; Atiyah and Macdonald Introduction to Commutative Algebra (1969) and Eisenbud Commutative Algebra with a View Toward Algebraic Geometry (1995) remain the standard modern textbooks.

The lineage from Gauss 1801 to Auslander-Buchsbaum 1959 spans 158 years and replaces the integer-specific fact "every positive integer factors uniquely into primes" with a structural framework in which the failure modes of factorisation are themselves algebraic invariants. Kummer's ideal numbers became Dedekind's ideals; Dedekind's class group became Iwasawa's $p$ -adic L-function obstruction; Hilbert's basis theorem became Buchberger's Gröbner-basis algorithm; the regular local ring criterion of Auslander-Buchsbaum identifies smooth points of varieties with rings of finite homological dimension. Each transformation replaces explicit calculation with a structural concept, and the cumulative effect is a framework in which problems from number theory, algebraic geometry, computer algebra, and homological algebra share a single vocabulary of rings, ideals, modules, and factorisations.

Contemporary developments extend the UFD-PID-Euclidean apparatus in several directions. The Iwasawa main conjecture (Mazur-Wiles 1984 Invent. Math. 76, Wiles 1990 Ann. Math. 131) connects class groups of cyclotomic fields with $p$ -adic L-functions, identifying analytic data with ideal-theoretic data. The Cohen-Lenstra heuristics 1983 Lecture Notes in Math. 1068 predict the statistical distribution of class groups of imaginary quadratic fields with a precision matching numerical experiments to seven decimal places. Lattice-based cryptography (NTRU, Ring-LWE) exploits the structure of polynomial rings $Z [x] / (f (x))$ where unique factorisation may or may not hold, with security analyses depending on the algebraic structure of the underlying ring. Derived algebraic geometry (Lurie 2009, Toën-Vezzosi 2008) replaces commutative rings with simplicial commutative rings and $E_{\infty}$ -ring spectra, generalising the UFD framework to a homotopical setting where unique factorisation becomes a derived-category statement about projective modules and Picard groups.

Bibliography Master

@book{Gauss1801,
  author = {Gauss, Carl Friedrich},
  title = {Disquisitiones Arithmeticae},
  publisher = {Fleischer},
  address = {Leipzig},
  year = {1801}
}

@article{Kummer1846,
  author = {Kummer, Ernst Eduard},
  title = {{\"U}ber die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume = {35},
  year = {1846},
  pages = {327--367}
}

@incollection{Dedekind1871,
  author = {Dedekind, Richard},
  title = {{\"U}ber die Theorie der ganzen algebraischen Zahlen},
  booktitle = {Vorlesungen {\"u}ber Zahlentheorie, 4. Auflage},
  editor = {Dirichlet, P. G. L.},
  publisher = {Vieweg},
  year = {1871},
  note = {Supplement XI}
}

@article{Hilbert1890,
  author = {Hilbert, David},
  title = {Ueber die Theorie der algebraischen Formen},
  journal = {Mathematische Annalen},
  volume = {36},
  year = {1890},
  pages = {473--534}
}

@article{Eisenstein1850,
  author = {Eisenstein, Gotthold},
  title = {{\"U}ber die Irreducibilit{\"a}t und einige andere Eigenschaften der Gleichung},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume = {39},
  year = {1850},
  pages = {160--179}
}

@article{Noether1921,
  author = {Noether, Emmy},
  title = {Idealtheorie in Ringbereichen},
  journal = {Mathematische Annalen},
  volume = {83},
  year = {1921},
  pages = {24--66}
}

@article{Krull1928,
  author = {Krull, Wolfgang},
  title = {Primidealketten in allgemeinen Ringbereichen},
  journal = {Sitzungsberichte der Heidelberger Akademie der Wissenschaften},
  volume = {7},
  year = {1928},
  pages = {11--29}
}

@article{Motzkin1949,
  author = {Motzkin, Theodore S.},
  title = {The Euclidean algorithm},
  journal = {Bulletin of the American Mathematical Society},
  volume = {55},
  year = {1949},
  pages = {1142--1146}
}

@article{AuslanderBuchsbaum1959,
  author = {Auslander, Maurice and Buchsbaum, David A.},
  title = {Unique factorization in regular local rings},
  journal = {Proceedings of the National Academy of Sciences of the USA},
  volume = {45},
  year = {1959},
  pages = {733--734}
}

@book{LangAlgebra,
  author = {Lang, Serge},
  title = {Algebra},
  publisher = {Springer},
  year = {2002},
  edition = {3rd},
  series = {GTM 211}
}

@book{AtiyahMacdonald,
  author = {Atiyah, Michael F. and Macdonald, Ian G.},
  title = {Introduction to Commutative Algebra},
  publisher = {Addison-Wesley},
  year = {1969}
}

@book{Eisenbud1995,
  author = {Eisenbud, David},
  title = {Commutative Algebra with a View Toward Algebraic Geometry},
  publisher = {Springer},
  year = {1995},
  series = {GTM 150}
}

@book{DummitFoote,
  author = {Dummit, David S. and Foote, Richard M.},
  title = {Abstract Algebra},
  publisher = {John Wiley \& Sons},
  year = {2004},
  edition = {3rd}
}

@book{Artin,
  author = {Artin, Michael},
  title = {Algebra},
  publisher = {Pearson},
  year = {2011},
  edition = {2nd}
}

Prerequisites

01.02.06
01.02.01
01.01.01

Tier anchors

beginner: Pinter A Book of Abstract Algebra Ch. 18-19; Allenby Rings, Fields and Groups Ch. 7
intermediate: Dummit-Foote 3e §8-9; Artin Algebra 2e §11.3, 12
master: Gauss 1801 Disquisitiones Arithmeticae; Kummer 1846 J. reine angew. Math. 35; Dedekind 1871 supplement XI; Hilbert 1890 Math. Ann. 36; Eisenstein 1850 J. reine angew. Math. 39; Motzkin 1949 Bull. AMS 55; Auslander-Buchsbaum 1959 Proc. Natl. Acad. Sci.; Lang Algebra 3e §II.5, V; Atiyah-Macdonald §1, 6, 10; Eisenbud §0-4

References

Gauss — Disquisitiones Arithmeticae · Leipzig: Fleischer, 1801
Kummer — Uber die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren · J. reine angew. Math. 35, 1846, 327-367
Dedekind — Uber die Theorie der ganzen algebraischen Zahlen · Supplement XI to Dirichlet's Vorlesungen 4e, 1871
Hilbert — Ueber die Theorie der algebraischen Formen · Math. Ann. 36, 1890, 473-534
Eisenstein — Uber die Irreducibilitat und einige andere Eigenschaften der Gleichung · J. reine angew. Math. 39, 1850, 160-179
Noether — Idealtheorie in Ringbereichen · Math. Ann. 83, 1921, 24-66
Krull — Primidealketten in allgemeinen Ringbereichen · S.-B. Heidelberg. Akad. Wiss. 7, 1928, 11-29
Motzkin — The Euclidean algorithm · Bull. AMS 55, 1949, 1142-1146
Auslander-Buchsbaum — Unique factorization in regular local rings · Proc. Natl. Acad. Sci. USA 45, 1959, 733-734
Lang — Algebra · GTM 211, 3e, §II.5, V
Atiyah-Macdonald — Introduction to Commutative Algebra · §1, 6, 10
Eisenbud — Commutative Algebra with a View Toward Algebraic Geometry · GTM 150, §0-4
Dummit-Foote — Abstract Algebra · 3e §8-9
Artin — Algebra · 2e §11.3, 12

Estimated time

beginner: 18m
intermediate: 45m
master: 80m