Teiji Takagi

How to use this course

Teiji Takagi was the first Japanese mathematician to change the course of world mathematics. In two long papers published in Tokyo in 1920 and 1922 he proved that the abelian extensions of a number field are exactly the class fields that Heinrich Weber and David Hilbert had defined, and he drew from this the general reciprocity laws that had been the goal of number theory since Gauss. He did this in isolation, cut off from Europe by the war. He also wrote the textbooks from which generations of Japanese students learned analysis and algebra, and trained the first Japanese school of algebraists.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 09 form one continuous course in class field theory, told in the order in which it was discovered. Lesson 10 is independent and can be read first by analysts.

Timeline

Gifu and Kyoto, 1875–1894

Teiji Takagi was born on 21 April 1875 in Kazuya, a village in the hills near Gifu in central Japan. His father kept the accounts of a farm, and the boy grew up there. His mother was a devout Buddhist who took him to the temple. Japan had opened to the West only two decades before his birth, and the Meiji government was building a national school system on European models.

Takagi went to primary school in his village and in 1886 entered the middle school in Gifu. There were no suitable mathematics books in Japanese, so he learned from English ones, including Isaac Todhunter’s algebra for beginners and a Victorian geometry text. Reading mathematics in a foreign language from the start shaped him. He later wrote textbooks so that Japanese students would not have to.

In 1891 he entered the Third High School in Kyoto, one of a handful of elite schools that prepared students for the Imperial University. He graduated in 1894.

Tokyo, Berlin, Göttingen, 1894–1901

Takagi entered the mathematics department of the Imperial University in Tokyo in 1894, at that time the only university in Japan. The department was small and its teaching was close to the European curriculum of a generation earlier. Takagi learned largely from books: George Salmon on algebraic curves, Joseph Serret’s Algèbre supérieure, and above all Heinrich Weber’s Lehrbuch der Algebra, which had just appeared. Weber’s book introduced him to Galois theory, algebraic numbers and elliptic functions, the three ingredients of his later work. He graduated in 1897 and began graduate study.

Germany

In 1898 the government sent him to Germany on a three-year scholarship. He spent the first half in Berlin, attending lectures by Georg Frobenius, Lazarus Fuchs and Hermann Amandus Schwarz. He found that he already knew much of what was taught. Midway through the scholarship he moved to Göttingen, where Hilbert had just published his Zahlbericht (1897) and was laying out a programme for class field theory.

Hilbert suggested a problem: prove Leopold Kronecker’s conjecture, his “dream of youth”, in the case of the Gaussian field \(\mathbb Q(i)\). The conjecture says that every abelian extension of an imaginary quadratic field is generated by special values of elliptic functions. Takagi solved the case of \(\mathbb Q(i)\) using the lemniscatic functions studied by Abel. Hilbert’s direct supervision was light, but Takagi later said that the Göttingen atmosphere had shaped his whole development. He saw how a modern school worked: lectures that reached the research frontier, a seminar, and a reading room with all the journals.

Professor in Tokyo, 1901–1914

Takagi returned to Japan in December 1901 and became assistant professor of algebra at Tokyo Imperial University. On 6 April 1902 he married Toshi Tani. His Göttingen work became his doctoral thesis, Über die im Bereich der rationalen complexen Zahlen Abel’schen Zahlkörper, published in the Journal of the College of Science of the university in 1903; the degree was conferred on 26 December 1903. In 1904 he was promoted to full professor, the post he held until 1936.

For the next decade he published little research. He was building a department. He wrote textbooks for schools and universities, beginning with A New Course of Arithmetic (1904), about 500 pages long, which constructed the real numbers from Dedekind cuts for Japanese readers. In the same period he published a short note giving a simple continuous function with no derivative anywhere, now called the Takagi function (Lesson 10). The note attracted no attention in Europe and the function was rediscovered several times.

When war broke out in 1914, German journals stopped arriving in Tokyo. Takagi later said that this isolation forced him to work alone, and that without it he might never have found his theory. He returned to the questions Hilbert had posed and, over the next four years, found that Hilbert’s restriction to unramified extensions had been an obstacle. Dropping it gave a complete theory.

Abelian extensions of ℚ

For the general reader. Some equations, such as \(x^5=1\), have roots that generate number systems with an especially simple symmetry group: one that is abelian, where the order of operations does not matter. A theorem of Kronecker and Weber says that over the rational numbers, every such system comes from roots of unity. Class field theory is the search for the analogue of this theorem over every number field.

Cyclotomic fields

Let \(\zeta_m=e^{2\pi i/m}\). The field \(\mathbb Q(\zeta_m)\) has degree \(\varphi(m)\) over \(\mathbb Q\), and the map \(a\mapsto(\sigma_a:\zeta_m\mapsto\zeta_m^a)\) is an isomorphism \((\mathbb Z/m\mathbb Z)^\times\to\mathrm{Gal}(\mathbb Q(\zeta_m)/\mathbb Q)\). A prime \(p\) ramifies in \(\mathbb Q(\zeta_m)\) only if \(p\mid m\).

Proof sketch. Modulo a prime \(\mathfrak P\) above \(p\), the Frobenius automorphism is \(x\mapsto x^p\), so it sends \(\zeta_m\) to \(\zeta_m^p\). Since \(x^m-1\) is separable mod \(p\), the reductions of the \(\zeta_m^a\) are distinct, and the automorphism that induces \(x\mapsto x^p\) on the residue field is \(\sigma_p\). The residue degree is the order of Frobenius.

By Galois theory, every subfield \(L\subseteq\mathbb Q(\zeta_m)\) corresponds to a subgroup \(H\le(\mathbb Z/m\mathbb Z)^\times\), and a prime \(p\nmid m\) splits completely in \(L\) exactly when \(p\bmod m\in H\). So the splitting of primes in \(L\) is governed by congruence conditions. That is the prototype of every statement in Lesson 07.

Kronecker announced it in 1853; Weber gave a proof in 1886 that contained a gap; Hilbert gave the first complete proof in 1896. Equivalently, the abelian extensions of \(\mathbb Q\) are generated by the values of \(e^{2\pi i z}\) at rational points \(z\). Hilbert’s twelfth problem (1900) asked for the analogous special functions over any number field. Kronecker’s Jugendtraum (Lesson 05) is the answer for imaginary quadratic fields.

Kronecker’s Jugendtraum and the thesis

For the general reader. Over the rational numbers, the exponential function produces all abelian extensions. Kronecker dreamed that over the Gaussian integers, or any imaginary quadratic field, elliptic functions would play the same role. Takagi’s thesis proved the dream for the Gaussian integers, using the function that measures arc length on the lemniscate, the figure-eight curve of Jacob Bernoulli.

Complex multiplication

The elliptic curve \(E:\ y^2=x^3-x\) has the extra automorphism \([i](x,y)=(-x,iy)\). Together with ordinary multiplication by integers this makes \(E\) a module over \(\mathbb Z[i]\): for each Gaussian integer \(\mu\) there is an endomorphism \([\mu]\) of degree \(N(\mu)=\mu\bar\mu\). Its kernel \(E[\mu]\) is a cyclic \(\mathbb Z[i]\)-module isomorphic to \(\mathbb Z[i]/(\mu)\). Over \(\mathbb C\), \(E\cong\mathbb C/\Lambda\) with \(\Lambda\) a multiple of \(\mathbb Z[i]\), and \(E[\mu]\) corresponds to \(\mu^{-1}\Lambda/\Lambda\). Abel had found in the 1820s that the division equations of the lemniscate are solvable by radicals, which is a shadow of the fact that their Galois groups are abelian.

The automorphisms of \(E\) are \(\pm1,\pm i\), and the function \(h(P)=x(P)^2\) is invariant under all of them. It plays the role of a Weber function.

In the language of Lesson 06, \(K_\mu\) is the ray class field of \(K\) modulo \(\mu\). Takagi’s thesis worked with the lemniscatic sine, which satisfies \((\mathrm{sl}')^2=1-\mathrm{sl}^4\), and its division values. The Galois action comes from the fact that \(\sigma\in\mathrm{Gal}\) acts on \(E[\mu]\) by \(\mathbb Z[i]\)-linear maps, hence by multiplication by elements of \((\mathbb Z[i]/\mu)^\times\). The hard direction, that every abelian extension appears, is where Takagi needed the analytic and arithmetic tools of Weber and Hilbert. His later general theory (Lesson 07) made this direction a consequence of a single existence theorem.

Ideal classes, ray classes, class fields

For the general reader. In many number systems unique factorization fails, and the class group measures by how much. Hilbert conjectured that each number field has a larger field, its class field, in which the failure is repaired and whose symmetry group is exactly the class group. Weber generalized class groups by adding congruence conditions. These were the objects Takagi had to work with.

Moduli and ray classes

Let \(K\) be a number field. A modulus \(\mathfrak m=\mathfrak m_0\mathfrak m_\infty\) is a non-zero ideal \(\mathfrak m_0\) of \(\mathcal O_K\) together with a set \(\mathfrak m_\infty\) of real embeddings. Let \(I^{\mathfrak m}\) be the group of fractional ideals coprime to \(\mathfrak m_0\), and \(P_{\mathfrak m,1}\) the subgroup of principal ideals \((\alpha)\) with \(\alpha\equiv1\pmod^{\times}\mathfrak m_0\) and \(\sigma(\alpha)>0\) for \(\sigma\in\mathfrak m_\infty\). The ray class group is \(\mathrm{Cl}_{\mathfrak m}=I^{\mathfrak m}/P_{\mathfrak m,1}\). It is finite, and there is an exact sequence

where \(\mathcal O^\times_{K,\mathfrak m,1}\) is the group of units that are \(\equiv1\bmod\mathfrak m_0\) and positive at \(\mathfrak m_\infty\). A congruence subgroup mod \(\mathfrak m\) is a subgroup \(H\) with \(P_{\mathfrak m,1}\subseteq H\subseteq I^{\mathfrak m}\). For \(K=\mathbb Q\) and \(\mathfrak m=m\infty\), \(\mathrm{Cl}_{\mathfrak m}\cong(\mathbb Z/m\mathbb Z)^\times\).

Class fields

Weber proved that a class field, if it exists, is unique. He could not prove existence in general. Hilbert conjectured (1898) that the case \(H=P_K\), \(\mathfrak m=1\), gives an unramified abelian extension with Galois group \(\mathrm{Cl}_K\), the Hilbert class field; Philipp Furtwängler proved this in 1907.

The theorems of 1920

For the general reader. Takagi proved that Weber’s class fields always exist, and that they are all the abelian extensions there are. Every abelian extension of a number field is therefore described by congruence conditions on primes, just as over the rationals. One theorem replaced half a century of special cases.

For a finite abelian extension \(L/K\) and a modulus \(\mathfrak m\) divisible by all ramified primes, put \(H_{L}(\mathfrak m)=P_{\mathfrak m,1}\,N_{L/K}(I_L^{\mathfrak m})\), where \(N_{L/K}\) is the ideal norm.

Takagi’s isomorphism was abstract: he showed that the two groups are isomorphic without exhibiting a natural map. Artin’s reciprocity law (Lesson 09) supplied the canonical map.

The two inequalities

Let \(L/K\) be abelian of degree \(n\) and \(\mathfrak m\) a modulus divisible by the ramified primes.

The analytic inequality \([I^{\mathfrak m}:H_L(\mathfrak m)]\le n\), due in essence to Weber. Write \(H=H_L(\mathfrak m)\) and \(h=[I^{\mathfrak m}:H]\). As \(s\to1^+\), the simple pole of the Dedekind zeta function of \(L\) gives \(\sum_{\mathfrak p\ \text{split}}N\mathfrak p^{-s}\sim\frac1n\log\frac1{s-1}\). Summing \(\log L(s,\chi)\) over the characters \(\chi\) of \(I^{\mathfrak m}/H\) gives \(h\sum_{\mathfrak p\in H}N\mathfrak p^{-s}+O(1)\). The trivial character contributes \(\log\frac1{s-1}\), and each non-trivial \(L(s,\chi)\) is holomorphic at \(s=1\), so its logarithm is bounded above. Every split prime is a norm, hence lies in \(H\). Therefore \(\frac hn\log\frac1{s-1}\le\log\frac1{s-1}+O(1)\), and \(h\le n\).

The algebraic inequality \([I^{\mathfrak m}:H_L(\mathfrak m)]\ge n\) for cyclic \(L/K\). This was Takagi’s decisive step. For \(L/K\) cyclic of prime degree \(\ell\) he counted the ambiguous ideal classes of \(L\), those fixed by the Galois group, and compared that count with the index of norms of units, by an extension of Gauss’s genus theory. The result, in modern terms, is the computation of a Herbrand quotient. An induction on degree extends it to all cyclic extensions.

Together the inequalities show \([I^{\mathfrak m}:H_L(\mathfrak m)]=[L:K]\) for cyclic \(L/K\), so \(L\) is the class field of \(H_L(\mathfrak m)\). Takagi then built general abelian extensions as composita of cyclic ones and proved existence by constructing, for each \(H\), a Kummer extension (after adjoining roots of unity) large enough to contain the desired class field and then cutting it down by Galois theory.

Reciprocity laws, 1922

For the general reader. Gauss’s law of quadratic reciprocity relates whether \(p\) is a square modulo \(q\) to whether \(q\) is a square modulo \(p\). Generalizing it to cubes, fifth powers and beyond occupied Gauss, Jacobi, Eisenstein, Kummer and Hilbert. In his second great paper, Takagi showed that class field theory produces these laws in every number field containing the relevant roots of unity.

Quadratic reciprocity from cyclotomy

Let \(p\) be an odd prime and \(p^*=(-1)^{(p-1)/2}p\). Then \(\mathbb Q(\sqrt{p^*})\) is the unique quadratic subfield of \(\mathbb Q(\zeta_p)\). For an odd prime \(q\ne p\),

The first equivalence is the decomposition law in a quadratic field; the second and third come from Lesson 04, since \(\sqrt{p^*}\) is fixed exactly by the squares in \((\mathbb Z/p)^\times\). This is quadratic reciprocity.

Power residues

Let \(\ell\) be a prime and \(K\) a number field containing the \(\ell\)-th roots of unity \(\mu_\ell\). For \(\alpha\in K^\times\) and a prime \(\mathfrak p\) of \(K\) coprime to \(\ell\alpha\), the \(\ell\)-th power residue symbol \(\left(\frac{\alpha}{\mathfrak p}\right)_\ell\in\mu_\ell\) is defined by

It equals 1 iff \(\alpha\) is an \(\ell\)-th power modulo \(\mathfrak p\), iff \(\mathfrak p\) splits completely in the Kummer extension \(L=K(\sqrt[\ell]{\alpha})\). More precisely, \(\left(\frac{\alpha}{\mathfrak p}\right)_\ell=\mathrm{Frob}_{\mathfrak p}(\sqrt[\ell]\alpha)/\sqrt[\ell]\alpha\). Extending multiplicatively in \(\mathfrak p\) gives a character \(\mathfrak a\mapsto\left(\frac{\alpha}{\mathfrak a}\right)_\ell\) on ideals coprime to \(\ell\alpha\).

In 1922 Takagi used his theory to prove the \(\ell\)-th power reciprocity law, for odd primes \(\ell\), in every number field containing \(\mu_\ell\), in the form Hilbert had formulated with norm residue symbols. Furtwängler had earlier proved important cases. Hilbert’s ninth problem (1900) asked for the most general reciprocity law in an arbitrary number field. Takagi’s work and Artin’s reciprocity law (1927) answered it for abelian extensions. The full answer, for the explicit symbols at primes over \(\ell\), came later through Hasse, Shafarevich and others.

From Takagi to Artin and Chevalley

For the general reader. Takagi showed that two groups always have the same shape. Emil Artin found the natural map between them in 1927, and Claude Chevalley rebuilt the whole subject in the 1930s around new objects, the ideles, which made the theory local and global at once. This lesson finishes the story Takagi started.

The Artin map

Let \(L/K\) be abelian and \(\mathfrak p\) a prime of \(K\) unramified in \(L\). There is a unique \(\mathrm{Frob}_{\mathfrak p}\in\mathrm{Gal}(L/K)\) with \(\mathrm{Frob}_{\mathfrak p}(x)\equiv x^{N\mathfrak p}\pmod{\mathfrak P}\) for all \(x\in\mathcal O_L\) and every \(\mathfrak P\mid\mathfrak p\). Extending multiplicatively gives the Artin map \(\psi_{L/K}: I^{\mathfrak m}\to\mathrm{Gal}(L/K)\) for any \(\mathfrak m\) divisible by the ramified primes.

Artin conjectured this in 1923 while studying his \(L\)-functions, and proved it in 1927 using Nikolai Chebotaryov’s density theorem and a reduction to cyclotomic extensions. For Kummer extensions the Artin map is the power residue symbol of Lesson 08, so Artin’s law contains all the power reciprocity laws. Philipp Furtwängler proved Hilbert’s principal ideal theorem in 1930, after Artin reduced it to a statement in group theory.

Ideles

Chevalley introduced the idele group \(\mathbb I_K=\prod'_v K_v^\times\) (restricted product over all places) in 1936. The idele class group \(C_K=\mathbb I_K/K^\times\) packages all ray class groups at once: for each modulus \(\mathfrak m\), \(\mathrm{Cl}_{\mathfrak m}\cong C_K/\overline{U_{\mathfrak m}}\) for a suitable open subgroup \(U_{\mathfrak m}\). Takagi’s theorems become: there is a continuous surjection \(C_K\to\mathrm{Gal}(K^{\mathrm{ab}}/K)\) whose kernel is the connected component of the identity, and finite abelian extensions correspond bijectively to open subgroups of finite index in \(C_K\), \(L\leftrightarrow K^\times N_{L/K}\mathbb I_L\). In 1940 Chevalley proved the algebraic inequality without analysis, so the whole theory became purely algebraic.

The Takagi function and a matrix theorem

For the general reader. In the nineteenth century, mathematicians were startled to learn that a curve can be continuous everywhere yet have a well-defined slope nowhere. Weierstrass’s example used trigonometric series. Takagi found a much simpler one: add up zigzags, each half as tall and twice as fast as the last. The result looks like a heap of whipped cream, and is also called the blancmange curve.

Let \(\phi(x)=\mathrm{dist}(x,\mathbb Z)\), the triangle wave of period 1 and height \(1/2\). Define

Takagi published this example in a two-page note in the first volume of the Proceedings of the Physico-Mathematical Society of Japan. It was rediscovered by van der Waerden in 1930 (in a base-10 version), by Hildebrandt in 1933, and by de Rham in 1957.

Proof. The terms are bounded by \(2^{-n-1}\), so the series converges uniformly and \(T\) is continuous. Fix \(x\), and for each \(n\) let \(u_n=k/2^n\le x<v_n=(k+1)/2^n\). For \(j\ge n\), \(2^ju_n\) and \(2^jv_n\) are integers, so the \(j\)-th term vanishes at both ends. For \(j<n\), the function \(2^{-j}\phi(2^j t)\) is linear on \([u_n,v_n]\) with slope \(\pm1\). Hence

an integer with the parity of \(n\). Consecutive quotients differ by \(\pm1\), so they cannot converge. But if \(T'(x)\) existed, these quotients over shrinking intervals containing \(x\) would converge to \(T'(x)\). So \(T'(x)\) does not exist.

Properties

From the definition, \(T(x)=\phi(x)+\tfrac12T(2x)\). Further facts: the maximum of \(T\) is \(2/3\) (Kahane, 1959), attained on a Cantor-like set of Hausdorff dimension \(1/2\) consisting of the \(x\) whose binary digits pair up as \(01\) or \(10\); and \(|T(x+h)-T(x)|=O(h\log(1/h))\), which is optimal.

Autonne–Takagi factorization

Takagi arrived at it while studying an interpolation problem related to work of Carathéodory, Fejér and Landau. The factorization is used today in numerical linear algebra and in physics, for instance to diagonalize Majorana mass matrices of neutrinos.

Recognition, 1920–1936

Takagi published Über eine Theorie des relativ Abel’schen Zahlkörpers in 1920, in the Journal of the College of Science of Tokyo Imperial University, a paper of 133 pages. That autumn he travelled to the International Congress of Mathematicians in Strasbourg and gave a short talk, in French, on his results. German mathematicians had been excluded from the congress, and those present did not grasp what he had done. While in Europe he visited Erich Hecke and Wilhelm Blaschke in Hamburg.

Recognition came through Germany after all. In 1922 Carl Ludwig Siegel, who had read the paper, persuaded Emil Artin to study it. Helmut Hasse lectured on class field theory in Kiel in 1923, presented Takagi’s theory to the German Mathematical Society in 1925, and in 1926 published the first part of his Klassenkörperbericht, which made the theory standard. Artin’s reciprocity law followed in 1927. In 1925 Hilbert wrote to Takagi asking whether his paper could be republished in Mathematische Annalen. The same year Takagi was elected to the Imperial Academy of Japan.

In 1932 Takagi was a vice-president of the International Congress in Zürich, and he was named to the committee that chose the first Fields Medals, awarded at Oslo in 1936 to Lars Ahlfors and Jesse Douglas. The other members were George Birkhoff, Élie Cartan, Constantin Carathéodory and Francesco Severi. He retired from the University of Tokyo in 1936.

Teacher of a nation, 1936–1960

Takagi’s influence in Japan came as much from his teaching and books as from his research. His students included Shokichi Iyanaga, who later worked with Artin in Hamburg; Kenjiro Shoda, who studied with Emmy Noether in Göttingen and brought her abstract algebra to Japan; and Sigekatu Kuroda, who married one of Takagi’s daughters. Through them and their students, among them Kenkichi Iwasawa, the Tokyo school of algebraic number theory grew directly from his work.

His books shaped mathematical education. Topics from the History of Nineteenth-Century Mathematics (1933), A Course of Analysis (1938) and Algebraic Theory of Numbers (1948) remained in print for decades, and were revised by him through several editions. The preface to the number theory book is the only place where he set out his overall view of class field theory after Artin.

In 1940 he received the Order of Culture, Japan’s highest honour for the arts and sciences. His house in Tokyo was destroyed by bombing in 1945, and he moved back to his native village before returning to Tokyo in 1947 to live with his eldest son. His wife died in 1952. Takagi died in Tokyo on 28 February 1960, aged 84. He was posthumously awarded the Order of the Rising Sun, First Class.

Legacy

• Class field theory: the Takagi existence theorem is the foundation of the theory, which in the twentieth century grew into local and idelic class field theory, Galois cohomology and the Langlands programme, its nonabelian successor.
• Japanese mathematics: he is regarded as the founder of the modern Japanese school; his students and their students include Iwasawa and many of the algebraists of the mid-century.
• Analysis: the Takagi function remains a standard example and an active research topic, and the Autonne–Takagi factorization is a standard tool of matrix theory.
• Honours: the Mathematical Society of Japan holds the Takagi Lectures in his name. His collected papers were edited by Iyanaga, Iwasawa, Kodaira and Yosida (Springer, 1990).

Connections

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

Teachers and advisors

• David Hilbert

Students

• Kenjiro Shoda
• Shokichi Iyanaga
• Sigekatu Kuroda
• Tadasi Nakayama

Influenced by

• Georg Frobenius 1898-1900
• Karl Weierstrass
• Leopold Kronecker
• Wilhelm Weber

Influenced

• Claude Chevalley
• Emil Artin
• Helmut Hasse
• Kenkichi Iwasawa

Colleagues and collaborators

• Carl Ludwig Siegel 1922

Family

• Sigekatu Kuroda

Also mentioned in this life

• Niels Henrik Abel
• Lars Ahlfors
• Georg Cantor
• Richard Dedekind
• Peter Gustav Lejeune Dirichlet
• Élie Cartan
• Évariste Galois
• Carl Friedrich Gauss
• Felix Hausdorff
• Erich Hecke
• Jacob Bernoulli
• Carl Gustav Jacob Jacobi
• Ernst Eduard Kummer
• Robert Langlands
• Emmy Noether
• Bartel van der Waerden

Fields

• algebraic number theory 9 lives
• class field theory 4 lives
• analysis 20 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Teiji Takagi, MacTutor History of Mathematics.
2. Teiji Takagi, Wikipedia.
3. National Diet Library, Portraits of Modern Japanese Historical Figures: Takagi Teiji.
4. Dictionary of Scientific Biography, Takagi, Teiji.
5. K. Miyake, Teiji Takagi, founder of the Japanese school of modern mathematics, Japanese Journal of Mathematics 2 (2007).
6. S. Iyanaga, Memories of Professor Teiji Takagi, in Class Field Theory: Its Centenary and Prospect, Math. Soc. Japan, 2001.

Mathematics

1. T. Takagi, Über die im Bereich der rationalen complexen Zahlen Abel’schen Zahlkörper, J. Coll. Sci. Imp. Univ. Tokyo 19 (1903).
2. T. Takagi, Über eine Theorie des relativ Abel’schen Zahlkörpers, J. Coll. Sci. Imp. Univ. Tokyo 41 (1920), 1–133.
3. T. Takagi, Über das Reciprocitätsgesetz in einem beliebigen algebraischen Zahlkörper, J. Coll. Sci. Imp. Univ. Tokyo 44 (1922), 1–50.
4. T. Takagi, Collected Papers, 2nd ed., Springer, 1990.
5. H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Jahresber. DMV 35 (1926).
6. D. A. Cox, Primes of the Form \(x^2+ny^2\), 2nd ed., Wiley, 2013 (the best introduction to the material of Lessons 05–09).
7. J. Neukirch, Algebraic Number Theory, Springer, 1999.
8. Timeline of class field theory, Wikipedia.
9. P. C. Allaart and K. Kawamura, The Takagi function: a survey, arXiv:1110.1691.
10. G. Frei, F. Lemmermeyer and P. Roquette (eds.), Emil Artin and Helmut Hasse: The Correspondence 1923–1958, Springer, 2014.