Bartel van der Waerden

How to use this course

Most mathematicians who learned algebra in the twentieth century learned it, directly or at one remove, from van der Waerden’s Moderne Algebra. The book turned the Göttingen and Hamburg lectures of Emmy Noether and Emil Artin into a single coherent subject built from groups, rings, fields and modules. That alone would secure his place. His research range was also unusually wide: enumerative and algebraic geometry, combinatorics, Galois theory, group theory, quantum mechanics, statistics, and, in the second half of his life, the history of ancient mathematics and astronomy.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons can be read in any order. Lesson 05 (arithmetic progressions) needs no background beyond induction and is a good place to start.

Timeline

Amsterdam, 1903–1924

Bartel Leendert van der Waerden was born in Amsterdam on 2 February 1903. His father, Theo van der Waerden, had trained as a civil engineer at Delft, taught mathematics and mechanics in secondary schools, and was a left-wing member of the Social Democratic Workers’ Party who sat in the provincial government of North Holland and later in the Dutch parliament. His two younger brothers became an engineer and a judge. The household was political, bookish and secular.

He attended the Hogere Burgerschool in Amsterdam and entered the University of Amsterdam in 1919, aged 16. Amsterdam was then an unusual place to learn mathematics. Gerrit Mannoury taught him topology and the philosophy of mathematics. Roland Weitzenböck, an Austrian specialist in invariant theory, introduced him to the algebra of forms. He heard L. E. J. Brouwer, the founder of intuitionism and of modern topology. His formal supervisor was the geometer Hendrik de Vries, whose interest in enumerative geometry set his first problem.

Enumerative geometry

Hermann Schubert’s nineteenth-century calculus counted geometric configurations: how many lines meet four given lines in space (two), how many conics touch five given conics (3264). Its methods worked, but no one could say why. Hilbert’s fifteenth problem asked for a rigorous foundation. Van der Waerden’s dissertation, De algebraiese grondslagen der meetkunde van het aantal, attacked exactly this question with the algebraic tools he was learning. He defended it on 24 March 1926. By then he had already spent most of a year in Göttingen, and his tools had changed.

Göttingen and Hamburg, 1924–1927

In the autumn of 1924 he went to Göttingen with a scholarship from the Rockefeller-funded International Education Board. He studied topology with Hellmuth Kneser and attended the lectures of Richard Courant and David Hilbert. The decisive encounter was with Emmy Noether. In the 1993 interview with Yvonne Dold-Samplonius he recalled that she showed him that the tools he needed for algebraic geometry already existed in her ideal theory. He became one of the core members of her circle, and his ideas on generic points and dimension (Lesson 06) were formed in that setting.

After military service at the naval base of Den Helder and his doctorate, he spent the academic year 1926–27 in Hamburg, then the liveliest algebra department in Europe. Emil Artin, Erich Hecke, Wilhelm Blaschke and Otto Schreier were all there. One day after lunch, with Artin and Schreier, he settled a problem he had heard in Holland, attributed to the Dutch mathematician Pierre Baudet: if the natural numbers are split into two classes, must one class contain arbitrarily long arithmetic progressions? Schreier suggested the right finite formulation, Artin suggested working with blocks and allowing any number of colours, and van der Waerden found the induction. The paper, Beweis einer Baudetschen Vermutung, appeared in the Nieuw Archief voor Wiskunde in 1927. Lesson 05 gives the proof.

In Hamburg he also attended Artin’s algebra course. Artin and he planned a joint textbook. Artin withdrew, and van der Waerden wrote it alone.

In February 1927 he habilitated at Göttingen with a thesis on the generalized theorem of Bézout, and from May 1927 to September 1928 he served as Courant’s assistant.

Groningen and Moderne Algebra, 1928–1931

In 1928, aged 25, he became professor at the University of Groningen. In the summer of 1929 he was a visiting professor in Göttingen, and that year he married Camilla Rellich, sister of the analyst Franz Rellich. They had three children; their daughter Helga later married the mathematician Walter Habicht.

The two volumes of Moderne Algebra appeared from Springer in 1930 and 1931 in the Grundlehren series. The title page said the book used lectures by Artin and Noether. The first volume covered groups, rings, fields, polynomial algebra, Galois theory (in Artin’s formulation) and ordered and real fields. The second covered elimination theory, ideal theory in the style of Noether, linear algebra over rings, algebras and representation theory.

The book’s novelty lay in its organization. Older algebra texts were about solving equations and computing invariants. Van der Waerden started from structures defined by axioms and derived the classical results as theorems about those structures. Readers learned algebra as the study of what can be proved from a short list of axioms. For two generations the book was the standard text, translated into many languages and revised through later editions, which appeared simply as Algebra. Its organization still governs graduate algebra courses.

In 1929 he also wrote Spinoranalyse, a short paper for physicists that set up the spinor notation still used in particle physics (Lesson 08). In May 1931 he accepted a chair of geometry at Leipzig.

The axiomatic method in algebra

For the general reader. Before Moderne Algebra, Galois theory was about permutations of roots of a given equation. Van der Waerden, following Artin, recast it as a correspondence between the subgroups of a group and the intermediate fields of a field extension. The new form is shorter, more general and easier to use. This lesson shows the shift in miniature.

Fields and extensions

For fields \(K\subseteq L\), \(L\) is a vector space over \(K\), and the degree \([L:K]\) is its dimension. Degrees multiply in towers: \([M:K]=[M:L][L:K]\). An extension is normal if every irreducible polynomial over \(K\) with one root in \(L\) splits into linear factors in \(L\), and separable if those roots are distinct. A finite extension that is normal and separable is Galois, and \(\mathrm{Gal}(L/K)\) is the group of field automorphisms of \(L\) fixing \(K\).

Proof sketch. The key step is \([L:L^G]\le|G|\). Let \(G=\{\sigma_1,\dots,\sigma_n\}\) and take \(m>n\) elements \(x_1,\dots,x_m\in L\). The \(n\times m\) linear system \(\sum_j\sigma_i(x_j)\,c_j=0\) has a non-zero solution \(c\in L^m\). Choose one with the fewest non-zero entries and normalize one entry to 1. Applying any \(\tau\in G\) permutes the equations, so \(\tau(c)\) is also a solution; \(c-\tau(c)\) has fewer non-zero entries, hence is zero. So \(c\in(L^G)^m\), and the equation for \(\sigma_i=\mathrm{id}\) is a linear dependence of the \(x_j\) over \(L^G\). The reverse inequality follows from Dedekind’s lemma that distinct automorphisms are linearly independent over \(L\). Separability and normality follow because each \(x\in L\) is a root of \(\prod(T-y)\), the product over the distinct elements \(y\) of the orbit \(Gx\), a polynomial with coefficients in \(L^G\) and distinct roots.

With Artin’s theorem in hand the proof is a few lines of bookkeeping. This is the form in which Moderne Algebra presented the theory, and the form in which it is still taught.

Arithmetic progressions in colourings

For the general reader. Paint every whole number red or blue, however you like. Van der Waerden proved that you cannot avoid long evenly spaced runs of a single colour: for any length you choose, one colour contains a progression like 7, 11, 15, 19 of that length. The theorem is one of the founding results of Ramsey theory, the study of order that must appear in any large enough structure.

The equivalence of the two forms uses a compactness argument: if each \(\{1,\dots,n\}\) had a bad colouring, a diagonal argument would produce a bad colouring of all of \(\mathbb N\).

The case \(k=3\), \(r=2\)

The proof idea is clearest in a small case. Split \(\{1,\dots,325\}\) into 65 consecutive blocks of 5. Each block has one of \(2^5=32\) colour patterns. Among the first 33 blocks two have the same pattern, say blocks \(B_i\) and \(B_{i+e}\) with \(1\le e\le 32\). The block \(B_{i+2e}\) also lies within the 65 blocks. Inside \(B_i\), among the first three positions two share a colour, say positions \(p\) and \(p+s\) with \(s\in\{1,2\}\); call it red. Position \(p+2s\) is still inside the block. If it is red we are done. Otherwise it is blue. Now look at position \(p+2s\) in block \(B_{i+2e}\):

• the red elements \(p\in B_i\) and \(p+s\in B_{i+e}\), with that position, form a progression with difference \(5e+s\);
• the blue elements \(p+2s\in B_i\) and \(p+2s\in B_{i+e}\), with that position, form a progression with difference \(5e\).

Whatever its colour, a monochromatic 3-term progression appears. So \(W(3;2)\le325\). The true value is 9 (Exercise 5.1).

The general induction

The general proof inducts on \(k\), and for each \(k\) proves a stronger statement by induction on the number of “focused” progressions: progressions of length \(k-1\), each monochromatic in a different colour, all pointing at the same next term. When \(r\) such progressions focus on one point, that point must extend one of them. The bounds grow like an Ackermann function. Saharon Shelah (1988) found a proof with primitive recursive bounds, and Timothy Gowers (2001), through his Fourier-analytic proof of Szemerédi’s theorem, obtained

Known exact values with two colours are \(W(3;2)=9\), \(W(4;2)=35\), \(W(5;2)=178\) and \(W(6;2)=1132\). Elwyn Berlekamp (1968) showed \(W(p+1;2)>p\,2^p\) for primes \(p\).

Descendants

Paul Erdős and Pál Turán asked whether a density version holds. Endre Szemerédi (1975) proved that every subset of \(\mathbb N\) of positive upper density contains arbitrarily long progressions; Hillel Furstenberg (1977) reproved it with ergodic theory. Ben Green and Terence Tao (2008) proved that the primes, which have density zero, contain arbitrarily long progressions.

Foundations of algebraic geometry

For the general reader. The Italian school of algebraic geometry, led by Guido Castelnuovo, Federigo Enriques and Francesco Severi, discovered deep results about curves and surfaces with geometric intuition that often outran its proofs. From 1927 van der Waerden set out to give the subject algebraic foundations, using the ideal theory of Noether. His series of papers Zur algebraischen Geometrie, begun in 1933, and his 1939 book were an essential step toward the rigorous geometry later built by André Weil, Oscar Zariski and Alexander Grothendieck.

Generic points and dimension

Fix a field \(K\) and a large algebraically closed field \(\Omega\supseteq K\) of infinite transcendence degree, a “universal domain.” An irreducible affine variety \(V\subseteq\Omega^n\) defined over \(K\) corresponds to a prime ideal \(\mathfrak p\subset K[x_1,\dots,x_n]\).

The idea, which van der Waerden credited to Noether, makes precise the Italian notion of a “general point” of a variety. A property holds at a generic point exactly when it holds on a dense open subset. Specialization \(\xi\to\eta\), meaning every polynomial relation over \(K\) satisfied by \(\xi\) is satisfied by \(\eta\), lets one pass from general to particular points.

Bézout’s theorem

The content of the theorem is the definition of multiplicity. In his 1927 papers van der Waerden defined the multiplicity of an intersection point by deforming the equations to general ones with the same degrees, whose \(\prod d_i\) common zeros are distinct, and counting how many of them specialize to the given point. He proved that this number is well defined. In the plane the multiplicity of two curves at a point \(P\) agrees with the length \(\dim_k\mathcal O_P/(f,g)\) of the local ring, the definition now in textbooks.

Chow forms

Let \(V\subseteq\mathbb P^n\) be irreducible of dimension \(d\) and degree \(g\). A general linear subspace of codimension \(d+1\) misses \(V\). The set of \((d+1)\)-tuples of hyperplanes \((u^{(0)},\dots,u^{(d)})\) whose intersection meets \(V\) is a hypersurface.

The paper was number IX of the series, written jointly with Wei-Liang Chow, who took his doctorate at Leipzig in 1936. Chow varieties were the first general construction of moduli spaces of subvarieties.

Galois groups of random polynomials

For the general reader. Write down a polynomial with random integer coefficients. Its roots will almost certainly be as unrelated as possible: every permutation of them is a symmetry. Van der Waerden proved this in 1934 and in 1936 conjectured how rare the exceptions are. The conjecture was proved in full by Manjul Bhargava in 2025.

Reduction modulo \(p\)

The permutation is the Frobenius element at a prime above \(p\), which acts on the roots of each \(g_i\) as \(x\mapsto x^p\) acts on \(\mathbb F_{p^{n_i}}\), a single \(n_i\)-cycle.

Van der Waerden’s argument: a polynomial whose reduction mod 2 is irreducible has an \(n\)-cycle; one whose reduction mod 3 is (irreducible of degree \(n-1\))·(linear) has an \((n-1)\)-cycle; one whose reduction mod 5 is (irreducible quadratic)·(distinct irreducibles of odd degree) has an element whose odd power is a transposition. A positive proportion of residue classes does each of these, and using more primes drives the proportion of exceptions to zero.

The conjectured exponent is sharp: the polynomials with \(a_n=0\) are reducible, and there are about \((2H+1)^{n-1}\) of them. Patrick Gallagher (1973) used the large sieve to get \(E_n(H)=O(H^{n-1/2}\log H)\). Bhargava’s proof of the full conjecture appeared in the Annals of Mathematics in 2025.

Spinors and the group-theoretic method

For the general reader. The electron has spin one-half: a full rotation changes the sign of its quantum state. Paul Dirac’s 1928 equation described it with four-component objects that do not transform like vectors. Van der Waerden, at Paul Ehrenfest’s request, explained in 1929 what these objects are and gave them a notation with dotted and undotted indices. The notation is still standard in supersymmetry and particle physics.

From \(\mathrm{SL}(2,\mathbb C)\) to Lorentz transformations

To a four-vector \(x^\mu=(x^0,x^1,x^2,x^3)\) assign the Hermitian matrix

where \(\sigma_0=I\) and \(\sigma_1,\sigma_2,\sigma_3\) are the Pauli matrices. For \(A\in\mathrm{SL}(2,\mathbb C)\) the map \(X\mapsto AXA^\dagger\) preserves Hermiticity and the determinant, hence the Minkowski norm.

Van der Waerden notation

An undotted spinor \(\psi_\alpha\) (\(\alpha=1,2\)) transforms by \(\psi_\alpha\mapsto A_\alpha{}^\beta\psi_\beta\); a dotted spinor \(\bar\chi_{\dot\alpha}\) transforms with the complex conjugate matrix \(\bar A\). These are the two inequivalent two-dimensional representations, the left- and right-handed Weyl spinors. Indices are raised and lowered with the antisymmetric symbol, \(\varepsilon^{12}=-\varepsilon^{21}=1\), \(\varepsilon_{12}=-1\) in one common convention:

A four-vector is a spinor with one undotted and one dotted index. A Dirac spinor is a pair \((\psi_\alpha,\bar\chi^{\dot\alpha})\). Every finite-dimensional irreducible representation of the Lorentz algebra is a symmetric tensor product of \(m\) undotted and \(n\) dotted spinors, labelled \((m/2,\,n/2)\).

The 1932 book

Die gruppentheoretische Methode in der Quantenmechanik (1932) presented representation theory, the rotation group and the symmetric group for physicists, alongside Hermann Weyl’s Gruppentheorie und Quantenmechanik (1928) and Eugene Wigner’s book of 1931. Many physicists of the time resisted these methods; van der Waerden’s book aimed to make them as elementary as possible.

Burnside groups and permanents

For the general reader. Two short contributions show van der Waerden’s talent for precise, answerable questions. With Friedrich Levi he computed exactly the largest finite group with a given number of generators in which every element cubed is the identity. And a one-line problem he posed in 1926, about averaging permutations, took 55 years to solve.

Burnside groups of exponent 3

William Burnside asked in 1902 whether a finitely generated group in which \(g^m=1\) for all \(g\) must be finite. The free Burnside group \(B(r,m)\) is the largest such group with \(r\) generators.

Burnside himself had shown finiteness for exponent 3. Later Sanov proved finiteness for exponent 4 (1940) and Marshall Hall for exponent 6 (1958). Pyotr Novikov and Sergei Adian showed in 1968 that \(B(r,m)\) is infinite for \(r\ge2\) and all odd \(m\) beyond a large bound.

The permanent conjecture

The permanent of an \(n\times n\) matrix is \(\operatorname{per}A=\sum_{\sigma\in S_n}\prod_i a_{i\sigma(i)}\), the determinant without signs. A matrix is doubly stochastic if its entries are non-negative and every row and column sums to 1. By Birkhoff’s theorem these are the convex combinations of permutation matrices. In 1926 van der Waerden asked in the Jahresbericht der DMV for the minimum of the permanent on this set.

Dmitry Falikman proved the inequality and Georgy Egorychev the full statement with the equality case, using the Alexandrov–Fenchel inequality for mixed volumes. Leonid Gurvits (2008) found a short proof through real-stable polynomials: for \(p(x)=\prod_i\sum_j a_{ij}x_j\), the coefficient of \(x_1\cdots x_n\) is \(\operatorname{per}A\), and an inductive inequality on capacities \(\inf_{x>0}p(x)/(x_1\cdots x_n)\) gives the bound.

Rank tests and statistics

For the general reader. To compare two groups of measurements without assuming the data are normally distributed, one can replace each value by its rank. In 1952 van der Waerden proposed replacing the ranks by the corresponding quantiles of the normal distribution. The resulting test loses almost nothing when the data are normal and can gain a great deal when they are not.

Let \(X_1,\dots,X_m\) and \(Y_1,\dots,Y_n\) be independent samples from continuous distributions \(F\) and \(G\), and let \(N=m+n\). Rank the pooled sample and let \(R_1,\dots,R_m\) be the ranks of the \(X\)’s. With \(\Phi\) the standard normal distribution function, define the scores \(a_j=\Phi^{-1}\!\big(j/(N+1)\big)\) and

The variance formula is the usual one for sampling \(m\) items without replacement from the \(N\) scores, whose mean is 0 by the symmetry \(a_{N+1-j}=-a_j\).

Chernoff and Savage proved this for the closely related Fisher–Yates–Terry–Hoeffding test with expected normal order statistics as scores; van der Waerden’s quantile scores have the same asymptotic behaviour. Van der Waerden also wrote Mathematische Statistik (1957), a textbook used widely in German-speaking countries.

Leipzig, 1931–1945

Van der Waerden arrived in Leipzig in May 1931 as professor of geometry. The faculty included Werner Heisenberg and Friedrich Hund, and he took part in their seminars on quantum mechanics. Leipzig was where he wrote most of Zur algebraischen Geometrie, supervised doctoral students including Herbert Seifert (1932), Wei-Liang Chow (1936) and Hans Richter (1936), and joined the editorial boards of Mathematische Annalen and the Grundlehren series in 1934.

Two years after he arrived, the Nazis took power. He kept his Dutch citizenship throughout. He declined an offer from the Institute for Advanced Study in Princeton in 1933. He wrote in support of Edmund Landau in 1934, and in 1935 he and Heisenberg protested at a faculty meeting against the dismissal of Jewish colleagues. His co-author Friedrich Levi lost his Leipzig post that year and emigrated. He continued to publish work by Jewish authors in the Annalen despite pressure. When Germany invaded the Netherlands in May 1940 he wrote to the authorities that he would not publicly take Germany’s side. His father had died in June 1940; his mother died in occupied Laren in 1942.

He nonetheless remained in his chair for the whole war, and that fact defined the next phase of his life. In December 1943 his Leipzig house was destroyed by bombing. The family moved to Dresden, where Franz Rellich was professor, then to Bischofswerda, and in 1945 to Austria. In the summer of 1945 they returned to the Netherlands.

Historians have weighed this period differently. Alexander Soifer’s The Scholar and the State (2015) is a critical reading; reviewers and other historians stress his protests and his refusal to change nationality. The documents support both a man who did not collaborate and one who chose to stay.

Amsterdam, Zürich and the history of science, 1945–1996

Back in the Netherlands he had no position. Hans Freudenthal arranged an offer from Utrecht, but the government refused to approve it because he had worked in Germany through the war. Freudenthal then found him work at Shell in Amsterdam, and he led the applied mathematics department of the new Mathematisch Centrum. In 1947–48 he was a visiting professor at Johns Hopkins University, which offered him a permanent post. He returned instead to a chair at the University of Amsterdam, which he held from 1948, and was elected to the Royal Netherlands Academy in 1949.

In 1951 he accepted the chair at the University of Zürich left vacant by the death of Rudolf Fueter. He stayed for the rest of his life, directed the mathematics institute until 1970, retired in 1972, and supervised more than 40 doctoral students there, among them the historians of mathematics Günther Frei and Erwin Neuenschwander and the mathematician and music theorist Guerino Mazzola.

Historian of science

His historical work began with Ontwakende wetenschap (1950), translated as Science Awakening (1954), a history of Egyptian, Babylonian and Greek mathematics written for mathematicians. A second volume on the birth of astronomy followed. In Geometry and Algebra in Ancient Civilizations (1983) he argued that Babylonian, Chinese, Indian and Greek mathematics shared a common Neolithic source, a thesis that remains contested. Sources of Quantum Mechanics (1967) collected and introduced the founding papers of 1917–1926, and A History of Algebra (1985) traced his own subject from al-Khwarizmi to Noether. From 1960 to 1993 he co-edited the Archive for History of Exact Sciences.

He died in Zürich on 12 January 1996, aged 92.

Legacy

• Algebra: Moderne Algebra fixed the structural form of the subject for the twentieth century.
• Geometry: generic points, multiplicities and Chow forms helped turn the Italian school’s intuition into rigorous algebraic geometry.
• Combinatorics and number theory: van der Waerden’s theorem opened Ramsey theory and additive combinatorics; the permanent and Galois-group conjectures drove decades of research.
• Physics: the dotted-index spinor notation is standard in quantum field theory.
• Honours: the Saxon Academy (1934), the Royal Netherlands Academy (1949), the Leopoldina (1960), honorary doctorates from Athens (1961) and Leipzig (1985), and the Pour le Mérite for Sciences and Arts (1973).

Connections

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

Teachers and advisors

• Emmy Noether 1924-1928
• Hendrik de Vries 1926

Students

• David van Dantzig 1931
• Erwin Neuenschwander
• Guerino Mazzola
• Günther Frei
• Hans Richter 1936
• Herbert Seifert 1932
• Wei-Liang Chow 1936

Influenced by

• Emil Artin 1926-1927
• Emmy Noether 1924-1933
• L. E. J. Brouwer
• Gerrit Mannoury
• Roland Weitzenböck

Influenced

• Oscar Zariski

Colleagues and collaborators

• Richard Courant 1927-1928
• Werner Heisenberg 1931-1945
• Friedrich Hund 1931-1945
• Friedrich Levi 1933
• Otto Schreier 1926-1927

Friends

• Hans Freudenthal 1945-1948

Family

• Franz Rellich
• Walter Habicht

Also mentioned in this life

• Muhammad ibn Musa al-Khwarizmi
• Pavel Alexandrov
• André Weil
• Richard Dedekind
• Paul Dirac
• Paul Ehrenfest
• Paul Erdős
• Joseph Fourier
• Georg Frobenius
• Évariste Galois
• Alexander Grothendieck
• Erich Hecke
• David Hilbert
• Hendrik Antoon Lorentz
• Hermann Minkowski
• Wolfgang Pauli
• Hermann Weyl

Fields

• abstract algebra 4 lives
• algebraic geometry 17 lives
• combinatorics 9 lives
• mathematical physics 45 lives
• statistics 7 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Bartel Leendert van der Waerden, MacTutor History of Mathematics.
2. Bartel Leendert van der Waerden, Wikipedia.
3. Y. Dold-Samplonius, Interview with Bartel Leendert van der Waerden, Notices of the AMS 44 (1997).
4. Waerden, Bartel van der, Neue Deutsche Biographie.
5. Waerden, Bartel Leendert van der, Historisches Lexikon der Schweiz.
6. Vita, Orden Pour le Mérite für Wissenschaften und Künste.
7. A. Soifer, The Scholar and the State: In Search of Van der Waerden, Birkhäuser, 2015.
8. N. Schappacher, review of Soifer.

Mathematics

1. B. L. van der Waerden, Moderne Algebra, 2 vols., Springer, 1930–31.
2. B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927) 212–216.
3. B. L. van der Waerden, How the proof of Baudet’s conjecture was found, in Studies in Pure Mathematics, Academic Press, 1971.
4. B. L. van der Waerden, Spinoranalyse, Nachr. Ges. Wiss. Göttingen (1929) 100–109.
5. B. L. van der Waerden, Die gruppentheoretische Methode in der Quantenmechanik, Springer, 1932.
6. F. Levi and B. L. van der Waerden, Über eine besondere Klasse von Gruppen, Abh. Math. Sem. Hamburg 9 (1933).
7. W.-L. Chow and B. L. van der Waerden, Zur algebraischen Geometrie IX, Math. Ann. 113 (1937).
8. B. L. van der Waerden, Einführung in die algebraische Geometrie, Springer, 1939.
9. M. Bhargava, Galois groups of random integer polynomials and van der Waerden’s Conjecture, Ann. of Math. 201 (2025).
10. W. T. Gowers, A new proof of Szemerédi’s theorem, GAFA 11 (2001).
11. L. Gurvits, Van der Waerden/Schrijver–Valiant like conjectures and stable polynomials, arXiv:0711.3496.
12. Van der Waerden theorem and Burnside group, Encyclopedia of Mathematics.
13. B. L. van der Waerden, Science Awakening, Noordhoff, 1954; A History of Algebra, Springer, 1985.