Richard Dedekind

How to use this course

Richard Dedekind spent almost his whole career at a technical college in his home town. From there he changed what mathematicians think a number is. He defined the real numbers by cuts, gave the first set-theoretic account of the natural numbers, invented ideals as sets of algebraic integers, and introduced words that every algebra student now uses: field, ring of integers, module, ideal. Emmy Noether liked to tell her students that everything was already in Dedekind.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 and 05 treat the foundations (1872 and 1888). Lessons 06–09 treat algebraic number theory and its function-field twin (1871–1882). Lesson 10 collects his late work on groups and lattices (1896–1900).

Timeline

Brunswick and Gauss, 1831–1854

Julius Wilhelm Richard Dedekind was born in Brunswick on 6 October 1831. He did not use his first two names as an adult. His father taught law at the Collegium Carolinum, a college between a school and a university, and his mother was the daughter of another Carolinum professor. He was the youngest of four children. His sister Julie became a writer, and his brother Adolf a senior judge.

At the Martino-Catharineum gymnasium he first preferred chemistry and physics. He turned to mathematics when he found the logic of physics unclear. At sixteen he entered the Collegium Carolinum, where he learned analytic geometry, calculus and the foundations of analysis, and in 1850 he moved to Göttingen.

A student of Gauss

Göttingen mathematics in 1850 was thin. Carl Friedrich Gauss was over seventy and taught little beyond elementary courses. Dedekind attended his lectures on least squares and later remembered them as the finest he ever heard. He joined, from its start, the new mathematics and physics seminar associated with the physicist Wilhelm Weber and the mathematician Moritz Stern, where he learned number theory. In 1852 he took his doctorate with a short thesis on Eulerian integrals. He was Gauss’s last doctoral student.

Dedekind knew he had been taught little modern mathematics, and he spent two years filling the gaps. He habilitated in 1854, the same year as Bernhard Riemann, and began lecturing as a Privatdozent on probability and geometry.

Dirichlet, Riemann and Galois, 1855–1858

When Gauss died in 1855, Göttingen called Peter Gustav Lejeune Dirichlet from Berlin. For Dedekind this was the turning point. He attended Dirichlet’s courses on number theory, potential theory and partial differential equations, and the two became friends. Historians see the relationship as the making of Dedekind. Dirichlet’s habit of replacing calculation with ideas became Dedekind’s own method.

He also grew close to Riemann, five years older and equally shy. The two attended each other’s lectures and talked mathematics constantly. Dedekind absorbed Riemann’s conceptual style in analysis and geometry, and after Riemann’s death in 1866 he spent years on his friend’s papers.

Galois theory

Évariste Galois’s papers had been published by Liouville in 1846, and few people understood them. Between 1856 and 1858 Dedekind lectured on Galois theory at Göttingen to a very small audience. His notes approach the Galois group through the field generated by the roots, close to the modern view. In 1857 he published a paper in Crelle’s Journal on higher congruences: the arithmetic of polynomials modulo a prime \(p\), with unique factorization into irreducibles. The idea of a field as a system of numbers closed under the four operations, which he later named a Körper, comes from this period.

Zürich and the idea of the cut, 1858–1862

In 1858 Dedekind was appointed professor at the Polytechnikum in Zürich, founded only three years earlier. He had to teach the differential calculus to engineers. He found he could not prove, from any clear definition of number, the basic fact that a bounded increasing sequence converges. Textbooks appealed to geometric intuition.

On 24 November 1858, by his own later account, he found the answer. A line is continuous because every division of its points into a left part and a right part is made by exactly one point. The rational numbers lack this property. A real number should therefore be defined as such a division of the rationals. He did not publish the idea for fourteen years. Lesson 04 develops it.

In 1859 he travelled with Riemann to Berlin, where Riemann was elected to the Academy, and met Karl Weierstrass, Ernst Kummer, Carl Borchardt and Leopold Kronecker. Kummer’s theory of ideal numbers for cyclotomic integers would become the starting point of his own number theory.

In 1862 the Collegium Carolinum became a Polytechnikum, and Dedekind accepted a chair there. He returned to Brunswick and never left. He declined later offers, including from Göttingen, and lived with his unmarried sister Julie until her death in 1914. He never married.

Cuts and the continuum

For the general reader. The square root of 2 is not a fraction, yet every fraction is either too small or too large to be it. Dedekind proposed that \(\sqrt2\) simply is this split of the fractions into “too small” and “too large”. Every real number is such a split, and the completeness of the real line follows by logic alone.

Cuts

In Stetigkeit und irrationale Zahlen (1872) Dedekind defines a cut \((A_1,A_2)\) of \(\mathbb Q\) as a partition into two non-empty classes such that every element of \(A_1\) is less than every element of \(A_2\). Each rational \(r\) produces two cuts, according to whether \(r\) is put in \(A_1\) or \(A_2\); he identifies them. A cut in which \(A_1\) has no largest element and \(A_2\) no smallest is produced by no rational. For each such cut he creates a new, irrational number.

Proof. No rational has square 2. For \(x>0\) put \(y=\dfrac{2x+2}{x+2}\). Then

If \(x\in A_1\) is positive, then \(x^2<2\), so \(y>x\) and \(y^2<2\): a larger element of \(A_1\). If \(x\in A_2\), then \(y<x\) and \(y^2>2\): a smaller element of \(A_2\). Non-positive elements of \(A_1\) are beaten by 1.

The real numbers

Let \(\mathbb R\) be the set of cuts, writing each cut as its lower class \(\alpha=A_1\) (with no largest element, by convention). Order cuts by inclusion. Addition is \(\alpha+\beta=\{a+b: a\in\alpha,\ b\in\beta\}\), and multiplication is defined first for positive cuts and then by signs. Dedekind remarked that this gives the first proof of identities such as \(\sqrt2\cdot\sqrt3=\sqrt6\).

Proof sketch. Let \(\gamma=\bigcup_{\alpha\in\mathfrak A_1}\alpha\), a union of lower classes. It is non-empty, it is not all of \(\mathbb Q\) because it lies inside any \(\beta\in\mathfrak A_2\), it is closed downward and has no largest element. So \(\gamma\) is a cut with \(\alpha\subseteq\gamma\subseteq\beta\). Uniqueness: two distinct cuts \(\gamma<\gamma'\) have a rational cut strictly between them, which must lie in one class and contradicts the bounds. The same union argument shows that every non-empty set of reals bounded above has a least upper bound. Dedekind then proved that a bounded monotone sequence converges, the fact that started him thinking in Zürich.

Georg Cantor published a construction by Cauchy sequences in the same year, and Charles Méray and Weierstrass had related ideas. Dedekind’s version is the most direct: a real number is a set of rationals.

Chains, infinity and the natural numbers

For the general reader. What is the number 3? Dedekind’s 1888 booklet Was sind und was sollen die Zahlen? answers with a structure: the natural numbers are any system with a first element and a “next” operation that never repeats and reaches everything. From this he derived proof by induction, definition by recursion and all of arithmetic. Along the way he gave the definition of an infinite set still used today.

Infinite sets

Galileo and Bolzano had noticed that an infinite set can match a part of itself. Dedekind turned the paradox into a definition. He tried to prove that an infinite system exists by pointing to the totality of his own possible thoughts. That argument does not survive the paradoxes of set theory, and Ernst Zermelo later replaced it with an axiom of infinity.

Chains

Let \(\varphi:S\to S\). A subset \(K\subseteq S\) is a chain (Kette) if \(\varphi(K)\subseteq K\). For \(A\subseteq S\), the chain of \(A\), written \(A_0\), is the intersection of all chains containing \(A\). It is the smallest chain containing \(A\).

Proof. \(A_0\cap\Sigma\) contains \(A\) and is a chain, since \(\varphi(A_0\cap\Sigma)\subseteq\varphi(A_0)\cap\Sigma\subseteq A_0\cap\Sigma\). By minimality, \(A_0\subseteq A_0\cap\Sigma\).

A system \(N\) is simply infinite if there are an injective \(\varphi:N\to N\) and an element \(1\in N\) such that \(1\notin\varphi(N)\) and \(N=\{1\}_0\). These are exactly the conditions Giuseppe Peano published in 1889 as axioms, independently. Dedekind calls the natural numbers the abstract form of any simply infinite system.

Uniqueness follows from induction. Existence is the subtle part, which Dedekind was the first to see needed proof: one builds, by induction on \(m\), maps defined on the initial segments \(\{1,\dots,m\}\) and shows they fit together. Taking \(\Omega\) to be a second simply infinite system yields:

Addition is then defined by recursion, \(m+1=\varphi(m)\) and \(m+\varphi(n)=\varphi(m+n)\), and multiplication likewise.

Countability of the algebraic numbers

In his letter to Cantor of 30 November 1873, Dedekind proved that the algebraic numbers can be listed in a sequence. Cantor printed the proof in his 1874 paper without naming him (Lesson 11).

Ideals and unique factorization

For the general reader. In the numbers \(a+b\sqrt{-5}\), the number 6 factors as \(2\cdot3\) and also as \((1+\sqrt{-5})(1-\sqrt{-5})\), and none of these pieces breaks down further. Ernst Kummer had rescued factorization in special cases by inventing “ideal numbers”. Dedekind replaced these phantoms by actual sets of numbers, which he called ideals, and proved that ideals always factor uniquely into prime ideals.

Number fields and integers

Dedekind defined a Körper (field) as a set of complex numbers closed under the four operations. A number field \(K\) is a field of finite degree \(n\) over \(\mathbb Q\). An algebraic integer is a root of a monic polynomial with integer coefficients, and the algebraic integers in \(K\) form a ring \(\mathcal O_K\), which he called the system of integers of \(K\). As an abelian group, \(\mathcal O_K\cong\mathbb Z^n\).

An ideal of \(\mathcal O_K\) is a non-empty subset closed under addition and under multiplication by elements of \(\mathcal O_K\). An ideal \(\mathfrak p\ne0,\mathcal O_K\) is prime if \(ab\in\mathfrak p\) implies \(a\in\mathfrak p\) or \(b\in\mathfrak p\). Dedekind defined the product \(\mathfrak a\mathfrak b\) as the ideal generated by all products \(ab\), and said \(\mathfrak a\) divides \(\mathfrak b\) if \(\mathfrak b=\mathfrak a\mathfrak c\) for some ideal \(\mathfrak c\).

Proof sketch (in the spirit of 1894)

1. Every non-zero ideal has finite index in \(\mathcal O_K\), since it contains a non-zero integer \(m\) and \(\mathcal O_K/m\mathcal O_K\) is finite. Hence ideals satisfy the ascending chain condition, and every non-zero prime ideal is maximal (a finite integral domain is a field).
2. For a prime \(\mathfrak p\), put \(\mathfrak p^{-1}=\{x\in K: x\mathfrak p\subseteq\mathcal O_K\}\). Then \(\mathfrak p\mathfrak p^{-1}=\mathcal O_K\). The key step uses that \(\mathcal O_K\) is integrally closed: if \(x\mathfrak a\subseteq\mathfrak a\) for a finitely generated \(\mathfrak a\ne0\), then \(x\) is an algebraic integer.
3. Existence: if some ideal had no factorization, choose a maximal such \(\mathfrak a\); it lies in a prime \(\mathfrak p\), and \(\mathfrak a\mathfrak p^{-1}\) strictly contains \(\mathfrak a\), so it factors, and so does \(\mathfrak a\).
4. Uniqueness: if \(\mathfrak p\) divides a product of primes, it contains one of them and equals it by maximality. Cancel with \(\mathfrak p^{-1}\) and repeat.

Dedekind reworked this proof three times, in 1871, 1877 and 1894, each time removing computations. In 1927 Noether showed that steps 1–2 contain exactly the right axioms: Noetherian, integrally closed, every non-zero prime maximal. Rings with these properties are now called Dedekind domains.

Norms and classes

The norm \(N(\mathfrak a)=|\mathcal O_K/\mathfrak a|\) is multiplicative, and \(N(\alpha\mathcal O_K)=|N_{K/\mathbb Q}(\alpha)|\). Two ideals are equivalent if \(\mathfrak a=\lambda\mathfrak b\) for some \(\lambda\in K^\times\). Dedekind proved that the number \(h_K\) of classes is finite. It measures the failure of unique factorization of elements: \(h_K=1\) exactly when \(\mathcal O_K\) is a unique factorization domain.

Splitting primes and the discriminant

For the general reader. An ordinary prime like 5 may stay prime in a bigger number system, or it may break into several prime ideals, sometimes with repeats. Dedekind found a simple rule: in most cases the prime splits exactly as a polynomial factors when its coefficients are read modulo 5. Repeated factors, called ramification, occur only at finitely many primes, those dividing the discriminant.

Let \(K=\mathbb Q(\theta)\) with \(\theta\in\mathcal O_K\), and let \(F\in\mathbb Z[x]\) be the minimal polynomial of \(\theta\). The index \(k=[\mathcal O_K:\mathbb Z[\theta]]\) is finite.

Proof idea. When \(p\nmid k\), the inclusion \(\mathbb Z[\theta]\subseteq\mathcal O_K\) induces \(\mathcal O_K/p\mathcal O_K\cong\mathbb Z[\theta]/p\cong\mathbb F_p[x]/(\bar F)\). The Chinese remainder theorem splits the right side as \(\prod_i\mathbb F_p[x]/(P_i^{e_i})\), and its prime ideals and their multiplicities can be read off. This is where Dedekind’s 1857 work on congruences modulo \(p\) returns. His 1878 paper also showed how to handle primes dividing the index, which are genuine exceptions.

The discriminant

If \(\omega_1,\dots,\omega_n\) is a \(\mathbb Z\)-basis of \(\mathcal O_K\) and \(\sigma_1,\dots,\sigma_n\) are the embeddings \(K\to\mathbb C\), the discriminant is \(d_K=\det(\sigma_i(\omega_j))^2\in\mathbb Z\). For \(\theta\) as above, \(\operatorname{disc}(F)=k^2\,d_K\).

Dedekind proved this by introducing the different (his Grundideal), an ideal whose norm is \(|d_K|\) and whose prime factors are exactly the ramified primes. The different remains a basic tool in algebraic number theory and arithmetic geometry.

The zeta function and the class number

For the general reader. Euler and Riemann studied the primes through a single function, the zeta function. Dedekind built the same function for every number field, summing over ideals instead of integers. Its behaviour near one point encodes the class number, the measure of how badly unique factorization fails.

In his supplements to Dirichlet’s lectures Dedekind defined, for a number field \(K\),

the sum over non-zero ideals of \(\mathcal O_K\), the product over non-zero prime ideals. The Euler product is unique factorization of ideals in analytic form. For \(K=\mathbb Q\) it is Riemann’s \(\zeta(s)\).

Proof idea. Fix an ideal class \(C\). Counting ideals \(\mathfrak a\in C\) with \(N(\mathfrak a)\le X\) amounts to counting lattice points of \(\mathcal O_K\) in a region of \(\mathbb R^{r_1}\times\mathbb C^{r_2}\), modulo units. The count is \(\kappa X+O(X^{1-1/n})\), where \(\kappa\) is the volume term \(2^{r_1}(2\pi)^{r_2}R_K/(w_K\sqrt{|d_K|})\), the same for every class. Summing over the \(h_K\) classes gives \(h_K\kappa X\) ideals of norm at most \(X\), and partial summation turns this into the residue. Erich Hecke later proved the meromorphic continuation to the whole plane and a functional equation.

Quadratic fields

For a quadratic field of discriminant \(d\), splitting of primes (Exercise 7.3) gives \(\zeta_K(s)=\zeta(s)L(s,\chi_d)\), where \(\chi_d\) is the Kronecker symbol. Since \(\zeta(s)\) has residue 1, the residue of \(\zeta_K\) is \(L(1,\chi_d)\). For imaginary \(K\), \(r_1=0\), \(r_2=1\), \(R_K=1\), and the formula becomes \(h_K=\dfrac{w_K\sqrt{|d|}}{2\pi}L(1,\chi_d)\). Evaluating \(L(1,\chi_d)\) gives, for \(d<-4\),

Dedekind–Weber: algebraic functions

For the general reader. Riemann studied algebraic curves through geometry and analysis on surfaces. In 1882 Dedekind and Heinrich Weber rebuilt the whole theory using only algebra, by treating functions on a curve exactly as Dedekind treated algebraic numbers. Points of the curve became prime ideals. The analogy between numbers and curves that they made precise drives much of modern number theory.

The dictionary

Let \(K\) be a finite extension of the rational function field \(\mathbb C(z)\). Dedekind and Weber replace \(\mathbb Z\) by \(\mathbb C[z]\) and \(\mathcal O_K\) by the integral closure of \(\mathbb C[z]\) in \(K\). Prime ideals over \(z-a\) correspond to the points of the Riemann surface lying over \(a\); ramification corresponds to branch points; the discriminant locates them. To include points at infinity they defined a point intrinsically, as a map from \(K\) to \(\mathbb C\cup\{\infty\}\) respecting the field operations, which is a place in modern language. A divisor (their Polygon) is a finite formal sum \(D=\sum n_PP\) of points, of degree \(\sum n_P\).

Their proof is purely algebraic, so it works over any algebraically closed field, a fact exploited fifty years later in the arithmetic of curves over finite fields. They defined the genus algebraically, via the different and the discriminant, and showed it agrees with Riemann’s topological genus. The degree formula for a cover then becomes:

Groups, lattices, group determinants

For the general reader. In his sixties Dedekind kept opening new subjects. He classified the groups in which every subgroup is “normal”, noticed that greatest common divisors and least common multiples obey a few laws that define what is now called a lattice, and passed to Frobenius the puzzle that started the theory of group characters.

Dedekind groups

A group in which every subgroup is normal is now called a Dedekind group; a non-abelian one is Hamiltonian, a name Dedekind chose because the quaternion group \(Q_8=\{\pm1,\pm i,\pm j,\pm k\}\) is the basic example.

Reinhold Baer extended the result to infinite groups in 1933.

Dualgruppen

In 1897 Dedekind abstracted two operations, \(\vee\) and \(\wedge\), which are commutative, associative and satisfy the absorption laws \(a\vee(a\wedge b)=a\) and \(a\wedge(a\vee b)=a\). He called such a structure a Dualgruppe; it is a lattice. Modules (additive subgroups of \(\mathbb C\)) under sum and intersection satisfy an extra law.

A lattice satisfying this is called modular. A modular lattice fails to be distributive exactly when it contains a copy of \(M_3\) (a bottom, a top and three incomparable middle elements). A lattice is modular exactly when it contains no copy of \(N_5\) (bottom, top, and a chain of two on one side with one element on the other). Dedekind proved the second criterion and, in 1900, that the modular lattice freely generated by three elements has exactly 28 elements. Garrett Birkhoff’s lattice theory of the 1930s grew from these papers.

The group determinant

For a finite group \(G\), introduce variables \(x_g\) and form the \(|G|\times|G|\) matrix with \((g,h)\) entry \(x_{gh^{-1}}\). Its determinant \(\Theta_G\) is the group determinant. Dedekind showed that for abelian \(G\) it factors into linear forms, one for each character \(\chi:G\to\mathbb C^\times\):

He found non-linear irreducible factors for \(S_3\) and \(Q_8\) and, in a letter of 1896, asked Frobenius how \(\Theta_G\) factors in general. Frobenius answered within months.

Brunswick, editions and Cantor, 1862–1894

The Brunswick Polytechnikum was a modest school, and Dedekind taught engineers analytic geometry and calculus for thirty years. He served as its rector from 1872 to 1875, during its reorganization. He turned down calls to larger universities and kept a small circle of correspondents. He was a good pianist and enjoyed accompanying singers.

Editing other people’s mathematics

Much of his best work appeared under other names. Dirichlet died in 1859, and Dedekind edited his number theory lectures as Vorlesungen über Zahlentheorie (1863). In the editions of 1871, 1879 and 1894 he added supplements that grew into a treatise on algebraic number theory, the first full presentation of ideals. After Riemann’s death in 1866, Dedekind and Heinrich Weber sorted his papers, and Weber’s edition of Riemann’s works (1876) included Dedekind’s biography of his friend. He also worked on the edition of Gauss’s collected works.

Cantor

In 1872, on a holiday in Switzerland, Dedekind met Georg Cantor at Gersau. Both had just published constructions of the real numbers. A correspondence followed. In late November 1873 Cantor asked whether the natural numbers could be matched with the real numbers. Dedekind replied on 30 November with a proof that the algebraic numbers are countable, and he simplified Cantor’s subsequent proof that the reals are not. Cantor’s 1874 paper contained both arguments with no mention of Dedekind. Dedekind’s letter was long thought lost; its recent recovery, reported in 2026, confirmed the extent of the borrowing that historians such as José Ferreirós had argued for. Dedekind stopped writing for some years. The two later resumed contact, and in the late 1890s Cantor sent Dedekind his letters on the paradoxes of set theory.

Leopold Kronecker developed a rival theory of divisors that avoided infinite sets. The contrast between Kronecker’s constructive approach and Dedekind’s set-theoretic one shaped arguments about foundations for decades.

Last years and legacy, 1894–1916

Dedekind retired on 1 April 1894 but kept teaching occasionally and kept publishing. His papers on groups, lattices and the group determinant date from his late sixties. He was elected to the Göttingen academy (1862), the Berlin Academy (1880), the academy in Rome and the Leopoldina, and the Paris Académie des Sciences (1900). He received honorary doctorates from Kristiania (Oslo), Zürich and Brunswick. His sister Julie died in 1914. Dedekind died in Brunswick on 12 February 1916, aged 84.

Legacy

• Foundations: Dedekind cuts and the definition of infinite sets; the recursion theorem and the categorical characterization of \(\mathbb N\), the model for axiomatic set theory and structuralism in mathematics.
• Algebra: the words and concepts of field, ring of integers, module and ideal; Noether’s abstract algebra was a deliberate generalization of his work, and she co-edited his collected works (1930–32).
• Number theory: unique factorization of ideals, the different and discriminant, the Dedekind zeta function. Hilbert’s 1897 Zahlbericht built on his foundations.
• Geometry: the Dedekind–Weber algebraic theory of curves, ancestor of the analogy between number fields and function fields used by Weil and Grothendieck.
• Other areas: lattice theory, Hamiltonian groups, and the prompt that led to Frobenius’s character theory.

Connections

People linked to Richard Dedekind across this collection. A name without a link has no page here yet. See Richard Dedekind in the network of all 337 lives →

Teachers and advisors

• Carl Friedrich Gauss 1852
• Peter Gustav Lejeune Dirichlet 1855-1859
• Wilhelm Weber

Influenced by

• Bernard Bolzano
• Ernst Kummer
• Eudoxus of Cnidus
• Peter Gustav Lejeune Dirichlet 1855-1859
• Évariste Galois

Influenced

• David Hilbert
• Emmy Noether
• Erich Hecke
• Gregorio Ricci-Curbastro

Colleagues and collaborators

• Felix Klein
• Georg Cantor 1872-1899
• Georg Frobenius 1896
• Gottlob Frege
• Rudolf Clausius 1858-1862
• Wilhelm Weber 1876-1882

Rivals

• Leopold Kronecker

Friends

• Bernhard Riemann 1854-1866
• Georg Cantor 1872-1899

Also mentioned in this life

• André Weil
• Augustin-Louis Cauchy
• Leonhard Euler
• Galileo Galilei
• Alexander Grothendieck
• Adolf Hurwitz
• Joseph Liouville
• Karl Weierstrass

Fields

• algebraic number theory 9 lives
• foundations of mathematics 10 lives
• abstract algebra 4 lives
• set theory 12 lives

On CoursesHub

• Cantor & Dedekind · History of Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Richard Dedekind, MacTutor History of Mathematics.
2. Richard Dedekind, Wikipedia.
3. Dedekind, Richard, Neue Deutsche Biographie.
4. Dedekind, Richard, Historisches Lexikon der Schweiz.
5. J. W. Richard Dedekind, Mathematics Genealogy Project.
6. J. Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Birkhäuser, 1999.
7. J. Howlett, The man who stole infinity, Quanta Magazine, 25 February 2026.

Mathematics

1. R. Dedekind, Stetigkeit und irrationale Zahlen, Vieweg, 1872; and Was sind und was sollen die Zahlen?, Vieweg, 1888. English in Essays on the Theory of Numbers, trans. W. W. Beman, 1901.
2. P. G. Lejeune Dirichlet and R. Dedekind, Vorlesungen über Zahlentheorie, 4th ed., Vieweg, 1894. English: Lectures on Number Theory, trans. J. Stillwell, AMS, 1999.
3. R. Dedekind, Theory of Algebraic Integers, trans. J. Stillwell, Cambridge University Press, 1996.
4. R. Dedekind and H. Weber, Theorie der algebraischen Functionen einer Veränderlichen, J. reine angew. Math. 92 (1882). English: Theory of Algebraic Functions of One Variable, trans. J. Stillwell, AMS, 2012.
5. R. Dedekind, Ueber die von drei Moduln erzeugte Dualgruppe, Math. Ann. 53 (1900).
6. R. Dedekind, Gesammelte mathematische Werke, ed. R. Fricke, E. Noether, Ø. Ore, 3 vols., Vieweg, 1930–32.
7. F. Q. Gouvêa and J. Webster, Dedekind on higher congruences and index divisors, 1871 and 1878, arXiv:2107.08905.
8. Frobenius determinant theorem and Dedekind zeta function, Wikipedia.
9. H. M. Edwards, The genesis of ideal theory, Archive for History of Exact Sciences 23 (1980).