Emil Artin

How to use this course

Emil Artin published comparatively little, and almost everything he published changed a field. His thesis began the arithmetic of curves over finite fields. His L-functions and his reciprocity law are the centre of class field theory and the starting point of the Langlands programme. He solved Hilbert’s seventeenth problem, invented the algebraic theory of braids, gave his name to Artinian rings, and rewrote Galois theory in the form taught today. He was also one of the great teachers of the century, in Hamburg and then in Princeton.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow his work roughly in order: function fields (1921), L-functions (1923–1930), reciprocity (1927), real fields (1926–1927), braids (1925, 1947), rings and Galois theory (1927–1944), and the primitive root conjecture (1927).

Timeline

Vienna and Reichenberg, 1898–1916

Emil Artin was born in Vienna on 3 March 1898. His mother, Emma Maria Artin, sang soubrette roles in operetta under the stage name Emma Clarus. His father, also Emil, had an Armenian family background and probably worked in the family art business. The father became seriously ill and died in 1906, when the boy was eight.

In 1907 his mother married Rudolf Hübner, a manufacturer in Reichenberg in northern Bohemia, now Liberec in the Czech Republic. Artin grew up there and attended the Realschule, a school oriented toward modern languages and science. His early marks in mathematics were unremarkable. He later wrote that his love of mathematics appeared only in his sixteenth year.

Two experiences of these years stayed with him. In 1912–13 he spent a school year with a French family near Paris, which gave him fluent French and, by his own account, one of the happiest years of his life. His marks improved sharply on his return. He also formed a close friendship with Arthur Beer, later an astronomer; the two boys shared an interest in the night sky. Astronomy, chemistry, biology and music remained lifelong interests. Artin left school in 1916 with a certificate of distinction.

War, Vienna and Leipzig, 1916–1921

Artin enrolled at the University of Vienna in October 1916, where he heard Philipp Furtwängler, a number theorist whose later work on the principal ideal theorem would meet Artin’s own (Lesson 06). After a semester he was called up. From June 1918 he served with the Austrian army on the Italian front. He returned briefly to Vienna after the armistice.

In January 1919 he moved to Leipzig and worked mainly with Gustav Herglotz, a versatile analyst and mathematical physicist. He also studied chemistry and physics. In June 1921 he received his doctorate with a thesis published in 1924 as Quadratische Körper im Gebiete der höheren Kongruenzen.

The thesis carried the arithmetic of quadratic number fields over to quadratic extensions of the field of rational functions over a finite field. Artin defined a zeta function for these fields, showed that it is a rational function, and observed that the analogue of the Riemann hypothesis could be checked by finite computation in each case. He checked it in many cases. The question he raised there occupied Hasse in the 1930s and André Weil in the 1940s. Lesson 04 tells the story.

Göttingen and Hamburg, 1921–1926

Artin spent the academic year 1921–22 at Göttingen, the centre of German mathematics, among Richard Courant, David Hilbert and Emmy Noether. He met Helmut Hasse there, who became his closest mathematical correspondent. Courant’s house concerts drew him into serious music. He played the flute and keyboard instruments and was devoted to Bach; later he owned a harpsichord and a clavichord.

On 1 October 1922 he took up an assistant’s post at the young University of Hamburg, founded in 1919. The department was led by Erich Hecke, a student of Hilbert and an expert on L-functions, and Wilhelm Blaschke, a geometer who had founded the department’s journal, the Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Nearly all of Artin’s major papers of the next fifteen years appeared there.

Artin habilitated on 24 July 1923 with a paper defining a new kind of L-series (Lesson 05). He became extraordinary professor on 1 April 1925 and full professor on 15 October 1926, at 28. Hecke and Blaschke pressed for the chair to keep him from accepting an offer elsewhere. In these four years he introduced the braid group (1925) and, with Otto Schreier, built the algebraic theory of real fields (1926). His algebra lectures of 1926, alongside Noether’s, became a principal source for Bartel van der Waerden’s Moderne Algebra (1930–31).

Zeta functions of function fields

For the general reader. The Riemann hypothesis, about the zeros of a function built from the primes, is the most famous open problem in mathematics. In his thesis Artin found a parallel world, built from polynomials over a finite field, where the same question can be asked and checked by counting. The analogue was proved there, first by Hasse for elliptic curves and then by Weil for all curves.

The polynomial ring as a model of the integers

Let \(A=\mathbb F_q[t]\). Every non-zero ideal is generated by a unique monic polynomial \(f\), with norm \(|f|=q^{\deg f}=\#A/(f)\). By analogy with \(\zeta(s)=\sum n^{-s}=\prod_p(1-p^{-s})^{-1}\), set

Artin studied the quadratic extensions \(K=\mathbb F_q(t)(\sqrt{D})\), with \(D\) a square-free polynomial and \(q\) odd. He developed ideal theory in the integral closure of \(A\) in \(K\), defined \(\zeta_K(s)\) as the sum over ideals of \(N(\mathfrak a)^{-s}\), and showed that in the variable \(u=q^{-s}\) it is a rational function with a functional equation. The numerator is a polynomial whose roots can be found in finite time.

The general statement

Friedrich Karl Schmidt extended rationality and the functional equation to all function fields in 1931, using the Riemann–Roch theorem. In geometric language, a function field of one variable over \(\mathbb F_q\) is the field of functions of a smooth projective curve \(C\). Let \(N_n=\#C(\mathbb F_{q^n})\).

Taking the logarithmic derivative gives \(N_n=q^n+1-\sum_i\alpha_i^n\). So the Riemann hypothesis gives the explicit estimate

For \(g=1\) this is Hasse’s theorem of 1933–36. Artin’s thesis contained the question; his friend answered it. Pierre Deligne proved the higher-dimensional analogue, conjectured by Weil, in 1974.

Artin L-functions

For the general reader. Dirichlet proved there are infinitely many primes in arithmetic progressions using functions built from characters of a finite abelian group. Artin generalized the construction to every finite group of symmetries of a number field, using matrices in place of numbers. His L-functions are still among the central objects of number theory, and his conjecture about them is still open.

Definition

Let \(L/K\) be a Galois extension of number fields with group \(G\), and \(\rho:G\to\mathrm{GL}(V)\) a finite-dimensional complex representation. For a prime \(\mathfrak p\) of \(K\), choose \(\mathfrak P\mid\mathfrak p\) in \(L\), with decomposition group \(D_{\mathfrak P}\), inertia group \(I_{\mathfrak P}\) and Frobenius \(\mathrm{Frob}_{\mathfrak P}\in D_{\mathfrak P}/I_{\mathfrak P}\). Then

which converges for \(\operatorname{Re}s>1\) and does not depend on the choice of \(\mathfrak P\). Artin’s 1923 paper treated unramified primes and left the ramified factors provisional; his 1930 paper gave the definition above, together with the conductor \(\mathfrak f(\rho)\) needed for the functional equation.

From (1) and (3) with \(H=1\), \(\zeta_L(s)=L(s,\operatorname{Ind}_1^G\mathbf 1)\), and the regular representation decomposes as \(\bigoplus_\chi\chi^{\chi(1)}\). Hence

the product over non-trivial irreducible characters of \(G\). For abelian \(G\) this recovers the classical factorization into Hecke L-functions.

Richard Brauer proved in 1947, with his induction theorem, that every \(L(s,\rho)\) has meromorphic continuation and a functional equation. The conjecture holds for one-dimensional \(\rho\) by Hecke’s theory, and for two-dimensional odd representations over \(\mathbb Q\) as a consequence of the proof of Serre’s modularity conjecture by Chandrashekhar Khare and Jean-Pierre Wintenberger. In general it remains open. The Langlands programme predicts that every Artin L-function is the L-function of an automorphic representation, which would imply the conjecture.

The general reciprocity law

For the general reader. Gauss’s quadratic reciprocity law says that whether \(p\) is a square modulo \(q\) is tied to whether \(q\) is a square modulo \(p\). A century of work produced cubic, quartic and higher laws, each with its own proof. In 1927 Artin found one law containing them all. It says that the way primes split in an abelian extension is controlled by a homomorphism, now called the Artin map.

Statement

Let \(L/K\) be an abelian extension of number fields with group \(G\), and let \(\mathfrak m\) be a modulus divisible by every ramified prime. For a prime \(\mathfrak p\nmid\mathfrak m\), the Frobenius \(\big(\frac{L/K}{\mathfrak p}\big)\in G\) is well defined because \(G\) is abelian. Extend multiplicatively to the group \(I_K^{\mathfrak m}\) of fractional ideals prime to \(\mathfrak m\).

Teiji Takagi had shown in 1920 that the two groups are isomorphic, without a canonical map. Artin conjectured the explicit isomorphism in his 1923 L-series paper and proved it in 1927. His proof used an idea from Nikolai Chebotaryov’s recent proof of the density theorem: adjoin auxiliary roots of unity to reduce to cyclic extensions where the claim can be checked. Artin named the result the general reciprocity law. Through Hasse’s reports it became the main theorem of class field theory.

The principal ideal theorem

Hilbert had conjectured that every ideal of \(K\) becomes principal in the Hilbert class field \(H\). In 1929 Artin showed, using reciprocity, that this is equivalent to a statement about groups: if \(G\) is a finite group with commutator subgroup \(G'\), the transfer \(G/G'\to G'/G''\) is trivial. Applied to \(G=\mathrm{Gal}(H_2/K)\), with \(H_2\) the Hilbert class field of \(H\), the transfer corresponds to extending ideals from \(K\) to \(H\). Furtwängler, his old teacher from Vienna, proved the group theoretic statement in 1930.

Real fields and Hilbert’s seventeenth problem

For the general reader. A polynomial that is never negative, like \(x^2+y^2\), may be a sum of squares. Hilbert knew that some non-negative polynomials are not sums of squares of polynomials, and in 1900 he asked whether they are always sums of squares of fractions. Artin proved that they are, after building with Otto Schreier an algebraic theory of which fields can be ordered like the real numbers.

Artin–Schreier theory (1926–1927)

A field \(F\) is formally real if \(-1\) is not a sum of squares in \(F\). An ordering is a subset \(P\) closed under addition and multiplication with \(P\cup(-P)=F\) and \(P\cap(-P)=\{0\}\). A field is real closed if it is formally real and has no proper formally real algebraic extension.

Hilbert’s seventeenth problem

Proof sketch. Let \(K=\mathbb R(x_1,\dots,x_n)\). If \(f\) is not a sum of squares in \(K\), part (1) gives an ordering of \(K\) in which \(f<0\). Let \(R\) be the real closure of \(K\) for this ordering. In \(R\) the point \((x_1,\dots,x_n)\) is a witness to the statement “there is \(a\) with \(f(a)<0\)”. Artin proved a specialization lemma: such a statement, with coefficients in \(\mathbb R\), true in a real closed extension of \(\mathbb R\), is already true in \(\mathbb R\). So some \(a\in\mathbb R^n\) has \(f(a)<0\), a contradiction. The specialization lemma is a special case of what Alfred Tarski later proved as the transfer principle for real closed fields.

Theodore Motzkin gave in 1967 the first explicit non-negative polynomial that is not a sum of squares of polynomials, so Artin’s fractions are necessary.

The theory of braids

For the general reader. Take \(n\) strings hanging between two bars and weave them without letting any string turn back. Two weavings count as the same if one can be deformed into the other. Artin saw in 1925 that these braids form a group, and found a short list of rules that describe it completely. The braid group now appears in knot theory, algebraic geometry, quantum physics and cryptography.

Definition

A geometric braid on \(n\) strands is a collection of \(n\) disjoint arcs in \(\mathbb R^2\times[0,1]\) joining \(n\) fixed points at height 0 to the same points at height 1, each arc meeting every horizontal plane once. Braids up to isotopy form a group \(B_n\) under stacking. Equivalently, \(B_n=\pi_1(\mathrm{UConf}_n(\mathbb C))\), the fundamental group of the space of unordered \(n\)-point subsets of the plane. Let \(\sigma_i\) be the braid in which strand \(i\) crosses over strand \(i+1\) and all others go straight.

Sending \(\sigma_i\) to the transposition \((i\ i{+}1)\) gives a surjection \(B_n\to S_n\), whose kernel is the pure braid group \(P_n\). Artin’s 1925 paper relied partly on pictures; in 1947 he gave a rigorous treatment in the Annals of Mathematics, including a normal form for pure braids (the “combing” of a braid) that solves the word problem. James Alexander had shown in 1923 that every knot or link is the closure of a braid, which is why braids became a tool in knot theory. Vaughan Jones discovered his knot polynomial in 1984 through representations of \(B_n\).

Artinian rings and Galois theory

For the general reader. Artin had a gift for finding the shortest route to a theorem. Two of his routes became the standard ones. He showed that the classification of algebras by Joseph Wedderburn needs only one finiteness condition, and he rebuilt Galois theory around linear algebra, so that the main theorem follows from counting dimensions.

Rings with minimum condition (1927)

Wedderburn had classified finite-dimensional semisimple algebras over a field in 1908. In a paper written in 1927 and published in 1928, Artin replaced finite dimension by the descending chain condition on one-sided ideals. A ring with this condition is now called Artinian, the counterpart of Noether’s ascending chain condition.

In 1944 Artin, Cecil Nesbitt and Robert Thrall gathered the theory in Rings with Minimum Condition.

Galois theory (1938–1942)

Artin’s lectures, printed as Galois Theory by Notre Dame in 1942, start from the independence of characters (Dedekind’s lemma): distinct homomorphisms \(\sigma_1,\dots,\sigma_n\) from a group into \(E^\times\) are linearly independent over \(E\).

Proof sketch. Let \(n=|G|\). If \(\alpha_1,\dots,\alpha_m\in E\) with \(m>n\), the system \(\sum_j x_j\,\sigma(\alpha_j)=0\) (\(\sigma\in G\)) has \(n\) equations and \(m\) unknowns, so it has a non-zero solution in \(E\). Choose one with fewest non-zero entries, normalize one entry to 1, and apply \(\tau\in G\): subtracting gives a shorter solution unless all \(x_j\in F\). Taking \(\sigma=1\) shows the \(\alpha_j\) are dependent over \(F\), so \([E:F]\le n\). Dedekind’s lemma gives \([E:F]\ge n\). The fundamental theorem follows by applying this to each subgroup.

The primitive root conjecture

For the general reader. The decimal expansion of \(1/7\) repeats with the longest possible period, six digits. This happens exactly when 10 is a primitive root modulo 7. Gauss asked whether there are infinitely many such primes. In 1927 Artin gave a precise prediction for how often it happens. Almost a century later, no single integer is known to be a primitive root for infinitely many primes.

Origin and statement

An entry of 27 September 1927 in Hasse’s mathematical diary records a density question “after an oral communication from Artin” of 13 September. For \(g\in\mathbb Q\setminus\{-1,0,1\}\), let \(N_g(x)\) be the number of primes \(p\le x\) for which \(g\) is a primitive root modulo \(p\).

The heuristic

\(g\) fails to be a primitive root mod \(p\) exactly when \(g\) is an \(\ell\)-th power mod \(p\) for some prime \(\ell\mid p-1\). That happens exactly when \(p\) splits completely in \(K_\ell=\mathbb Q(\zeta_\ell,g^{1/\ell})\), a field of degree \(\ell(\ell-1)\) for generic \(g\). By Chebotaryov the proportion of such \(p\) is \(1/(\ell(\ell-1))\). Treating different \(\ell\) as independent gives the product \(A\). The independence fails when the fields \(K_\ell\) are not linearly disjoint, which is why the density must be corrected when the square-free part of \(g\) is \(\equiv1\pmod 4\): then \(\sqrt g\in\mathbb Q(\zeta_{|D|})\). Numerical work by Derrick and Emma Lehmer exposed the discrepancy. Artin had found the correction in a 1958 letter to them, and Hans Heilbronn published the corrected form.

What is known

Christopher Hooley proved the conjecture, with the corrected density, in 1967, assuming the generalized Riemann hypothesis for the fields \(\mathbb Q(\zeta_k,g^{1/k})\). Unconditionally, Roger Heath-Brown showed in 1986, building on work of Rajiv Gupta and Ram Murty, that among any three multiplicatively independent primes at least one is a primitive root for infinitely many \(p\). His result does not say which one.

Hamburg, exile and America, 1926–1958

Hamburg in the late 1920s was one of the best places in the world to learn algebra and number theory. Artin taught algebra, mechanics and relativity, and he was dean of the faculty in 1931–32. His doctoral students there included Käte Hey, who wrote in 1927 on zeta functions of algebras, Max Zorn, and Hans Zassenhaus. His long correspondence with Hasse ran from 1923 into the 1930s and is a primary source for the making of class field theory. In 1932 he shared the Ackermann–Teubner Memorial Prize with Emmy Noether.

On 15 August 1929 he married Natascha Jasny, a mathematics student born in 1909. Their children were Karin (born 1933), Michael (born 1934), who became an algebraic geometer at MIT, and Tom (born 1938 in Bloomington).

The Nazi years

Artin was not dismissed in 1933, unlike his Jewish colleagues. His name appears among the signatories of the November 1933 declaration of German professors in support of Hitler; one account reports that he later said it had been added without his knowledge. He was openly hostile to the regime. Natascha’s father came from a Jewish family, so under Nazi law she counted as a “Mischling”. In 1937 a decree removing civil servants married to persons of Jewish descent was applied to him. He was ordered into retirement on 27 July 1937, effective 31 October. The family left for the United States in October 1937.

America

With help from Richard Courant and Solomon Lefschetz, Artin found a post at the University of Notre Dame for 1937–38. From 1938 to 1946 he was professor at Indiana University in Bloomington. There and at Notre Dame he wrote the lectures that became Galois Theory (1942), and with George Whaples he characterized global fields axiomatically by the product formula for valuations (1945). He became a US citizen in 1946.

In the autumn of 1946 he moved to Princeton, where he held the Dod chair from 1948 and the Fine chair from 1953. Princeton was the most productive period of his teaching. He supervised about eighteen doctorates there, among them John Tate, Serge Lang and Timothy O’Meara. Bernard Dwork also wrote his thesis under Artin, receiving his degree from Columbia. Tate married Artin’s daughter Karin. The Artin–Tate seminar on class field theory in 1951–52 recast the theory in the language of group cohomology; its notes circulated for years and appeared as the book Class Field Theory. Artin’s Geometric Algebra (1957) treated symplectic and orthogonal groups with the economy typical of his style. He also taught the honours calculus course for first-year students, which he took as seriously as his graduate seminars.

Return, death and legacy, 1958–1962

Hamburg granted Artin the status of a retired full professor in 1953 as restitution. He was a guest professor at Göttingen in the winter of 1956–57. Princeton’s retirement age was approaching, and Hasse and Emanuel Sperner arranged a new chair for him at Hamburg in 1958. He resigned from Princeton in 1959. He and Natascha divorced in 1959; she later married the composer Mark Brunswick. In his last years Artin shared his life with the mathematician Hel Braun. He regained German citizenship in 1961.

In June 1962 the University of Clermont-Ferrand gave him an honorary doctorate on the tercentenary of Pascal’s death. He died of heart failure at his home in Hamburg on 20 December 1962, aged 64. His urn was later moved to Vienna. Richard Brauer, Claude Chevalley, Henri Cartan and Hans Zassenhaus wrote memorial articles. In 2005 the University of Hamburg named a lecture hall after him.

Legacy

• Number theory: the Artin map and Artin L-functions are the foundation of class field theory; the Artin conjecture and its generalization by Robert Langlands organize much of modern research.
• Arithmetic geometry: his thesis began the study of zeta functions of varieties over finite fields, completed by Weil and Deligne.
• Algebra: Artinian rings, Artin–Schreier theory, real algebraic geometry and the modern presentation of Galois theory.
• Topology: the braid groups, now also studied as Artin groups in the wider family he began.
• People: through Tate, Lang, Dwork, Zassenhaus and many others, his style of clear, economical exposition became a norm. His son Michael Artin continued in algebraic geometry.

Sources and further reading

Biography

1. Emil Artin, Wikipedia (with extensive references).
2. R. Brauer, Emil Artin, Bull. Amer. Math. Soc. 73 (1967), 27–43, with bibliography.
3. Dictionary of Scientific Biography, Artin, Emil (via MacTutor).
4. Neue Deutsche Biographie, Artin, Emil.
5. K. Reich, Emil Artin, in the University of Hamburg history of its mathematics department, Hamburg University Press.
6. G. Frei, F. Lemmermeyer, P. Roquette (eds.), Emil Artin and Helmut Hasse: The Correspondence 1923–1958, Springer, 2014; German edition of the letters 1923–1934, Universitätsverlag Göttingen, 2008.
7. H. Zassenhaus, Emil Artin, his life and his work, Notre Dame J. Formal Logic 5 (1964).

Mathematics

1. E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen I, II, Math. Z. 19 (1924).
2. E. Artin, Über eine neue Art von L-Reihen, Abh. Math. Sem. Hamburg 3 (1923), 89–108.
3. E. Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Sem. Hamburg 5 (1927), 353–363.
4. E. Artin, Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Hamburg 5 (1927), 100–115.
5. E. Artin, O. Schreier, Algebraische Konstruktion reeller Körper, Abh. Math. Sem. Hamburg 5 (1927), 85–99.
6. E. Artin, Zur Theorie der hyperkomplexen Zahlen, Abh. Math. Sem. Hamburg 5 (1928), 251–260.
7. E. Artin, Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren, Abh. Math. Sem. Hamburg 8 (1930), 292–306.
8. E. Artin, Theory of braids, Ann. of Math. 48 (1947), 101–126.
9. E. Artin, Galois Theory, Notre Dame, 1942; Geometric Algebra, Interscience, 1957.
10. E. Artin, J. Tate, Class Field Theory, Benjamin, 1967; AMS Chelsea reprint.
11. S. Lang, J. Tate (eds.), The Collected Papers of Emil Artin, Addison-Wesley, 1965.
12. P. Moree, Artin’s primitive root conjecture: a survey, arXiv:math/0412262.
13. C. Kassel, V. Turaev, Braid Groups, Springer, 2008.