Richard Brauer

How to use this course

Richard Brauer is one of the few mathematicians whose name is attached to a group, an algebra, a character theory and a famous conjecture, all in different corners of algebra. Number theorists meet him through the Brauer group and the Brauer–Siegel theorem. Group theorists meet him through modular characters, blocks and the strategy that led to the classification of finite simple groups. Combinatorialists and physicists meet him through the Brauer algebra. This course follows all these threads and the life that connected them.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Most exercises use the small groups \(S_3\), \(A_4\) and \(A_5\), so you can check every claim by hand. The lessons roughly follow the order of Brauer’s work: Berlin (1926), Königsberg (1927–1933), Princeton and Toronto (1935–1948), and the group-theoretic programme of the 1950s.

Timeline

Berlin, 1901–1925

Richard Dagobert Brauer was born on 10 February 1901 in Charlottenburg, then a separate town on the western edge of Berlin. His family was Jewish and comfortably off: his father Max ran a wholesale leather business. Richard was the youngest of three children. His brother Alfred, seven years older, was the person who first drew him to science and mathematics. Alfred was wounded as a soldier in the First World War and later became a mathematician too. Their sister Alice was murdered in the Holocaust.

From 1907 to 1918 Richard attended the Kaiser-Friedrich-Schule in Charlottenburg. He later said his interest in mathematics owed more to Alfred than to his teachers. He graduated in September 1918 and was drafted into civilian war service in Berlin, from which he was released when the war ended in November.

From engineering to theory

As a boy he wanted to be an inventor. In February 1919 he enrolled at the Technische Hochschule in Charlottenburg, but after one term he moved to the University of Berlin, having found that his gift lay in theory. Apart from a summer term in Freiburg in 1920, he studied entirely in Berlin. His teachers there formed a remarkable list: Ludwig Bieberbach, Constantin Carathéodory, Albert Einstein, Konrad Knopp, Richard von Mises, Max Planck, Erhard Schmidt, Issai Schur and Gábor Szegő.

Two of them shaped him. Erhard Schmidt lectured without notes, and Brauer admired the clarity of his improvised exposition. Issai Schur ran a seminar and problem sessions that drew Brauer into algebra and the representation theory of groups. In 1921 Schur posed a problem that Richard and Alfred solved together; Heinz Hopf found a solution at the same time. The work became Richard’s first paper. In Schur’s number theory course he met Ilse Karger, also a mathematics student. They married in September 1925.

Königsberg, 1925–1933

Schur suggested the thesis problem: find an algebraic route to the characters of the irreducible representations of the real orthogonal groups. Hermann Weyl had just obtained them by integrating over the group; Brauer wanted a method that used only algebra. The dissertation, Über die Darstellung der Drehungsgruppe durch Gruppen linearer Substitutionen, was accepted with distinction on 16 March 1926.

Before the degree was formally awarded he had already left Berlin. In autumn 1925 he became assistant to Konrad Knopp at the University of Königsberg in East Prussia. In 1927 he obtained the habilitation and became a Privatdozent. Both of his sons, George Ulrich (born 1927) and Fred Günter (born 1932), were born in Königsberg, and both later became mathematicians.

Algebras

In Königsberg Brauer turned to the structure of finite-dimensional algebras, the field Emmy Noether was then reshaping in Göttingen. In 1927 they published a joint paper on splitting fields of simple algebras. In papers of 1928–1932, notably Über Systeme hyperkomplexer Zahlen (1929), Brauer developed factor sets and showed that the similarity classes of central simple algebras over a field form a group. The name “Brauer group” was given later, by others, and against his own wishes. In late 1931 Brauer, Helmut Hasse and Noether completed the proof that every central division algebra over a number field is cyclic (published 1932). In 1932 Brauer also published Über die algebraische Struktur von Schiefkörpern, his systematic account of division algebras. Lesson 05 covers this mathematics.

By 1933 he was regarded as one of the best young algebraists in Germany. That spring the new civil-service law removed Jewish academics from their posts, and Brauer was dismissed.

Exile: Lexington and Princeton, 1933–1935

The Emergency Committee in Aid of Displaced Foreign Scholars, set up in New York to place academics expelled from Germany, found Brauer a one-year visiting professorship at the University of Kentucky. He arrived in Lexington in November 1933. Ilse and the two boys followed about three months later. Brauer was thirty-two. He had to begin again in a new language, and the move cost him his place in the German mathematical world, but it also brought him into contact with American and Canadian mathematics at a time of rapid growth.

In 1934 Hermann Weyl invited him to the Institute for Advanced Study in Princeton as his assistant. Brauer admired Weyl deeply, and the year was productive. Together they wrote Spinors in n dimensions (1935), which constructed the spin representations of the orthogonal groups in every dimension through Clifford algebras. Brauer and Nathan Jacobson prepared notes of Weyl’s lectures on the structure and representation of continuous groups.

Emmy Noether, also exiled and teaching at Bryn Mawr, recommended Brauer for a post at the University of Toronto. She died in April 1935, a few months before he took up the position. In the autumn of 1935 Brauer became an assistant professor at Toronto, where he would spend the next thirteen years. Jonathan Alperin later called them the most productive period of his life.

Rotation groups, spinors and the Brauer algebra

For the general reader. The rotations of space form a group, and physicists and mathematicians want to know every way this group can act on vectors, tensors and more exotic objects. Brauer’s thesis found these actions algebraically. With Weyl he constructed the strange “half” representations called spinors in every dimension, and in 1937 he found the algebra of diagrams that controls how rotations act on tensors.

Tensors and invariants

Let \(V=\mathbb C^N\) with the symmetric bilinear form \(\langle x,y\rangle=\sum x_iy_i\), and let \(O(N)\) be the group preserving it. Schur had shown that the operators on \(V^{\otimes k}\) commuting with \(\mathrm{GL}(N)\) are exactly the linear combinations of permutations of the tensor factors. For \(O(N)\) there are more commuting operators, because the form itself is invariant. On \(V\otimes V\), the map \(x\otimes y\mapsto\langle x,y\rangle\sum_i e_i\otimes e_i\) commutes with \(O(N)\) but is not a permutation.

The Brauer algebra (1937)

A Brauer diagram on \(k+k\) points is a perfect matching of \(2k\) points arranged in two rows of \(k\). There are \((2k-1)!!=1\cdot3\cdot5\cdots(2k-1)\) of them. Diagrams are multiplied by stacking; each closed loop formed in the middle is removed and replaced by a factor \(\delta\). The span of the diagrams is the Brauer algebra \(B_k(\delta)\). The \(k!\) diagrams in which every strand joins top to bottom form a copy of the symmetric group \(S_k\).

The proof combines Weyl’s first fundamental theorem of invariant theory for \(O(N)\), which says every invariant of vectors is a polynomial in their inner products, with the double commutant theorem. A closed loop acts as \(\sum_i\langle e_i,e_i\rangle=N\), which is why \(\delta=N\).

Spinors

The Clifford algebra \(\mathrm{Cl}_N\) is generated by \(e_1,\dots,e_N\) subject to \(e_ie_j+e_je_i=2\delta_{ij}\). It has dimension \(2^N\), with basis the products \(e_{i_1}\cdots e_{i_r}\), \(i_1<\cdots<i_r\).

Élie Cartan had found spinors in 1913 by Lie algebra methods. Brauer and Weyl’s 1935 paper gave an explicit construction for all \(N\) through Clifford algebras, the approach that later became standard in physics for Dirac spinors in any dimension.

Central simple algebras and the Brauer group

For the general reader. The quaternions extend the complex numbers by allowing multiplication that does not commute. Over any field one can ask how many such number systems exist. Brauer found that they form a group in a natural way, and this group became one of the main invariants of a field, and later of a geometric space.

Definitions

A central simple algebra over a field \(K\) is a finite-dimensional \(K\)-algebra \(A\) with centre \(K\) and no two-sided ideals other than \(0\) and \(A\). By Wedderburn’s theorem, \(A\cong M_n(D)\) for a unique central division algebra \(D\), and \(\dim_KA\) is a square \(d^2\); \(d\) is the degree. Call \(A\) and \(B\) similar if they have isomorphic underlying division algebras.

Examples: \(\mathrm{Br}(K)=0\) when \(K\) is algebraically closed; \(\mathrm{Br}(\mathbb F_q)=0\) by Wedderburn’s little theorem that finite division rings are commutative; \(\mathrm{Br}(\mathbb R)\cong\mathbb Z/2\), generated by Hamilton’s quaternions, by Frobenius’s theorem.

Splitting fields, period and index

A field extension \(L/K\) splits \(A\) if \(A\otimes_KL\cong M_d(L)\). Every central division algebra of degree \(d\) contains a maximal subfield of degree \(d\) over \(K\), which splits it, and one can choose it separable. For a Galois extension \(L/K\) with group \(G\), the algebras split by \(L\) are crossed products, built from a factor set \(\{a_{\sigma,\tau}\}\subset L^\times\); in modern language

The index of \(A\) is the degree of its division algebra; the period is the order of \([A]\) in \(\mathrm{Br}(K)\).

Quaternion algebras and the local–global principle

For \(a,b\in K^\times\) (characteristic not 2), the quaternion algebra \((a,b)_K\) has basis \(1,i,j,ij\) with \(i^2=a\), \(j^2=b\), \(ij=-ji\). It is either split or a division algebra, and it splits exactly when \(ax^2+by^2=1\) has a solution in \(K\). Over \(\mathbb Q_p\) or \(\mathbb R\) this is decided by the Hilbert symbol \((a,b)_v=\pm1\).

For quaternion algebras over \(\mathbb Q\) this, with Hilbert’s product formula \(\prod_v(a,b)_v=1\), gives: the set of places where \((a,b)_{\mathbb Q}\) is a division algebra is finite and of even size, and it determines the algebra up to isomorphism.

Modular representations and Brauer characters

For the general reader. A group can act on vector spaces over any field. Over the complex numbers the theory, created by Georg Frobenius and Schur, is clean: every representation breaks into irreducible pieces, and characters tell you everything. Over a field of characteristic \(p\) dividing the order of the group this fails. Brauer spent much of his career building the theory for that case, and found that it is still controlled by complex-valued functions, now called Brauer characters.

Setting

Let \(G\) be finite, \(p\) a prime, and \(k\) an algebraically closed field of characteristic \(p\). An element \(g\in G\) is \(p\)-regular if its order is prime to \(p\). When \(p\mid|G|\), Maschke’s theorem fails: \(kG\) is not semisimple, and modules need not be direct sums of simple ones.

If \(M\) is a \(kG\)-module and \(g\) is \(p\)-regular, the eigenvalues of \(g\) on \(M\) are roots of unity of order prime to \(p\) in \(k\). These lift uniquely to complex roots of unity of the same order (fix once and for all a compatible isomorphism of the groups of such roots). The Brauer character \(\varphi_M(g)\) is the sum of the lifted eigenvalues. It is defined only on \(p\)-regular elements.

Idea of (2) and (3). Take a complex representation, realize it over a ring of algebraic integers localized at a prime above \(p\), and reduce modulo that prime. The reduction depends on the chosen lattice, but its composition factors, and hence its Brauer character, do not. Projective \(kG\)-modules lift to characteristic 0, and the lift of the projective cover \(P_\varphi\) has character \(\Phi_\varphi=\sum_\chi d_{\chi\varphi}\chi\) (Brauer reciprocity). Reducing \(\Phi_\varphi\) gives \(c_{\varphi\psi}=\sum_\chi d_{\chi\varphi}d_{\chi\psi}\).

Blocks, defect groups and the height zero conjecture

For the general reader. Brauer found that the irreducible characters of a finite group fall into families, called blocks, according to their behaviour modulo a prime \(p\). Each block carries a small \(p\)-subgroup, its defect group, which measures how far the block is from the easy semisimple case. Much of the deep structure of a finite group can be read off from its blocks and their defect groups. Some of Brauer’s questions about blocks were answered only in the 2020s.

Blocks

Let \(\mathcal O\) be a complete discrete valuation ring of characteristic 0 with residue field \(k\) of characteristic \(p\), large enough to contain the needed roots of unity. The group ring \(\mathcal OG\) decomposes uniquely as a product of indecomposable two-sided ideals \(B_1\times\cdots\times B_r\), the blocks. Each ordinary irreducible character and each simple \(kG\)-module belongs to exactly one block.

For an irreducible character \(\chi\) and a conjugacy class \(K\) with representative \(g_K\), the central character \(\omega_\chi(\hat K)=|K|\chi(g_K)/\chi(1)\) is an algebraic integer.

Defect and height

Write \(|G|_p=p^a\). For a block \(B\), the smallest \(p\)-part of \(\chi(1)\) for \(\chi\in B\) is \(p^{a-d}\); the integer \(d\) is the defect of \(B\). Each \(\chi\in B\) then has \(\chi(1)_p=p^{a-d+h}\) with \(h\geq0\), its height. Brauer attached to \(B\) a defect group \(D\), a \(p\)-subgroup of \(G\) determined up to conjugacy, with \(|D|=p^d\). The principal block, containing the trivial character, has the Sylow \(p\)-subgroups as defect groups. A block of defect zero contains a single character \(\chi\), with \(p^a\mid\chi(1)\).

Brauer’s second and third main theorems describe how characters of \(G\) restrict to centralizers of \(p\)-elements and identify the principal block under this correspondence. Together they make blocks a local theory: global information about \(G\) is recovered from normalizers and centralizers of \(p\)-subgroups.

Two conjectures

The “if” direction was completed by Radha Kessar and Gunter Malle in 2013, building on a reduction to simple groups by Thomas Berger and Reinhard Knörr. The “only if” direction was completed in 2024 by Malle, Gabriel Navarro, A. A. Schaeffer Fry and Pham Huu Tiep, after Lucas Ruhstorfer had settled \(p=2\).

Brauer’s \(k(B)\) conjecture asserts that the number \(k(B)\) of ordinary irreducible characters in \(B\) is at most \(|D|\). It is known for \(p\)-solvable groups and remains open in general.

Induced characters and Artin L-functions

For the general reader. Emil Artin attached an L-function, a cousin of the Riemann zeta function, to each representation of a Galois group. He conjectured that these functions behave well on the whole complex plane, but could prove it only in special cases. Brauer showed that every character of a finite group is an integer combination of characters induced from very simple subgroups. This one theorem of pure group theory gave the analytic continuation of all Artin L-functions to meromorphic functions.

A finite group \(E\) is \(p\)-elementary if \(E\cong C\times P\) with \(C\) cyclic of order prime to \(p\) and \(P\) a \(p\)-group; it is elementary if it is \(p\)-elementary for some prime \(p\).

Proof sketch. The \(\mathbb Z\)-span \(I\) of the induced characters is an ideal in the character ring \(R(G)\), by Frobenius reciprocity: \(\psi\cdot\mathrm{Ind}\lambda=\mathrm{Ind}(\psi|_E\cdot\lambda)\). So it suffices to show the trivial character \(1_G\) lies in \(I\). For each prime \(p\), one constructs an element of \(I\) that is congruent to \(1_G\) modulo a power of \(p\) large enough to absorb \(|G|_p\), using the \(p\)-elementary subgroups. The proof of Brauer and John Tate (1955) does this by an argument with integer-valued class functions. Combining the primes gives \(1_G\in I\). The characterization follows: if \(\theta|_E\) is virtual for all \(E\), then \(\theta=\theta\cdot1_G=\sum n_E\,\mathrm{Ind}(\theta|_E\lambda_E)\) is virtual.

Artin L-functions

Let \(L/K\) be a Galois extension of number fields with group \(G\), and \(\rho\) a representation of \(G\) with character \(\chi\). Artin defined \(L(s,\chi,L/K)\) as an Euler product over the primes of \(K\). Two formal properties hold: \(L(s,\chi_1+\chi_2)=L(s,\chi_1)L(s,\chi_2)\), and for an intermediate field \(F\) with \(H=\mathrm{Gal}(L/F)\) and a character \(\psi\) of \(H\),

For a one-dimensional \(\psi\), Artin reciprocity identifies \(L(s,\psi,L/F)\) with a Hecke L-function of \(F\), which is entire for \(\psi\neq1\) and has a simple pole at \(s=1\) for \(\psi=1\).

By the induction theorem, \(L(s,\chi)=\prod L(s,\lambda_E)^{n_E}\) is a product of integer powers of Hecke L-functions, hence meromorphic. Since some \(n_E\) may be negative, poles cannot be excluded. Artin’s conjecture, that \(L(s,\chi)\) is entire for nontrivial irreducible \(\chi\), remains open in general.

The Brauer–Siegel theorem

For the general reader. The class number of a number field measures how badly unique factorization fails. Carl Ludwig Siegel showed in 1935 that for imaginary quadratic fields the class number grows roughly like the square root of the discriminant. In 1947 Brauer extended this to very general families of number fields, where the class number is coupled with another invariant, the regulator.

For a number field \(K\) of degree \(n\), write \(d_K\) for the discriminant, \(h_K\) for the class number, \(R_K\) for the regulator, \(w_K\) for the number of roots of unity, and \(r_1,r_2\) for the numbers of real and complex places. The analytic class number formula reads

Sketch. By the class number formula, the theorem is equivalent to \(\log\operatorname{Res}_{s=1}\zeta_K=o(\log|d_K|)\), since the factors \(2^{r_1}(2\pi)^{r_2}/w_K\) contribute only \(O(n)\) to the logarithm, which is \(o(\log|d_K|)\) under the hypothesis. The upper bound for the residue is elementary. The lower bound is the hard part. Siegel’s method needs a field in which the zeta function has no real zero very close to 1. Brauer’s contribution was to use his results on Artin L-functions (Lesson 08) to factor \(\zeta_K\) and transfer such information between fields, which is where the Galois or bounded-degree hypothesis enters.

The constants are ineffective. The proof cannot rule out one exceptional field with a real zero near 1 (a Siegel zero), and it gives no way to compute where the asymptotic takes hold. Effective lower bounds for imaginary quadratic class numbers came only with the work of Dorian Goldfeld, Benedict Gross and Don Zagier in the 1980s.

Involutions and the road to classification

For the general reader. Finite simple groups are the atoms from which all finite groups are built. In 1954 Brauer proposed a strategy for finding them all: look at an element of order 2 and the subgroup of elements commuting with it. He proved that this subgroup almost determines the whole group. Thirty years and thousands of pages later, his strategy had become the backbone of the classification of finite simple groups.

An involution is an element of order 2. By Cauchy’s theorem every group of even order has one. William Burnside had conjectured that every non-abelian finite simple group has even order; if so, every such group has involutions to work with.

Proof of the bound

Let \(G\) be non-abelian simple, of order \(g\), with involution set \(I\) of size \(m\). The class of \(t\) has \(g/n\) elements, so \(m\geq g/n\). For \(z\in G\) let \(a(z)\) be the number of pairs \((x,y)\in I\times I\) with \(xy=z\). Then \(\sum_za(z)=m^2\) and \(a(1)=m\).

If \(xy=z\) with \(x,y\in I\), then \(xzx^{-1}=x(xy)x=yx=z^{-1}\): \(x\) inverts \(z\). The elements inverting \(z\) form a coset of \(C_G(z)\) or the empty set, and \(y=xz\) is determined by \(x\). So \(a(z)\le|C_G(z)|\). Averaging over the \(g-1\) non-identity elements, some \(z\neq1\) has

Since \(G\) is simple and non-abelian, \(Z(G)=1\) and \(C_G(z)\) is a proper subgroup. Its index is \(r\le g\big/\big(\tfrac g{n^2}-\tfrac1n\big)=\tfrac{n^2g}{g-n}=n^2+\tfrac{n^3}{g-n}\). If \(g>n^3+n\), the last term is below 1, so \(r\le n^2\). Then \(G\) acts on the \(r\) cosets of \(C_G(z)\); the kernel is a proper normal subgroup, hence trivial, so \(G\) embeds in \(S_r\) and \(g\le(n^2)!\). If \(g\le n^3+n\), the bound holds anyway for \(n\geq2\).

The programme

Brauer announced the strategy at the International Congress in Amsterdam in 1954: given a candidate centralizer \(H\), determine all simple groups with an involution centralizer isomorphic to \(H\). With Fowler he characterized the groups \(\mathrm{PSL}(2,q)\) in this way. With Michio Suzuki he proved that a group whose Sylow 2-subgroups are generalized quaternion is never simple; Suzuki’s and Brauer’s character-theoretic methods became standard. In 1963 Walter Feit and John Thompson proved Burnside’s conjecture that groups of odd order are solvable, so every non-abelian simple group has involutions, and Brauer’s approach applied to all of them. Zvonimir Janko’s discovery of the first new sporadic group in 1965 came from exactly such a centralizer analysis. The classification was announced in the early 1980s, and its last gap, the quasithin case, was closed by Michael Aschbacher and Stephen Smith in 2004.

Toronto, Ann Arbor, Harvard, 1935–1971

At Toronto Brauer found colleagues, students and the freedom to work. His first doctoral student, Cecil Nesbitt, wrote with him the 1937 paper that founded the theory of blocks. Other Toronto students included Richard Bruck, Stephen Jennings, Nathan Mendelsohn, Ralph Stanton and Robert Steinberg, who later became a leading figure in the theory of algebraic groups. A series of papers from 1941 to 1947 built up modular character theory and the theory of blocks.

In 1941–42 a Guggenheim Fellowship took him to the University of Wisconsin. He visited Emil Artin, another refugee, then teaching at Bloomington, Indiana, and spent time at the Institute for Advanced Study. The conversations with Artin led to the 1947 paper on Artin L-series, which won the American Mathematical Society’s Cole Prize in Algebra in 1949. He was elected a Fellow of the Royal Society of Canada in 1945.

In the summer of 1948 he gave the AMS Colloquium Lectures in Madison, and later that year moved to a chair at the University of Michigan, where his students included Kenneth Fowler and Donald J. Lewis. In 1952 he moved to Harvard. There he trained a new generation of finite group theorists, among them Paul Fong, Morton Harris, I. Martin Isaacs, Henry Leonard, Donald Passman, Walter Reynolds, Louis Solomon, David Wales, Harold Ward and Warren Wong.

Honours accumulated: election to the American Academy of Arts and Sciences (1954) and the National Academy of Sciences (1955), the presidency of the American Mathematical Society (1957–58), honorary membership of the London Mathematical Society (1963), and membership of the Göttingen Academy of Sciences (1964). His colleagues remembered a gentle, modest man who worked steadily and generously and who was always ready to share problems.

Last years and legacy, 1971–1977

Brauer retired from Harvard in 1971 but kept working. On 21 May 1971 President Nixon presented him with the National Medal of Science for 1970, cited for his work on conjectures of Dickson, Cartan, Maschke and Artin. He was elected to the American Philosophical Society in 1974, and in 1975 he published a paper introducing the notion of resolvent degree for polynomial equations. In March 1977 he was diagnosed with aplastic anaemia. He died of a general infection on 17 April 1977 in Boston, with Ilse beside him. His collected papers, edited by Paul Fong and Warren Wong, appeared in three volumes from MIT Press in 1980.

Legacy

• The Brauer group grew far beyond fields. Maurice Auslander and Oscar Goldman defined it for commutative rings in 1960, and Alexander Grothendieck for schemes in the 1960s. The Brauer–Manin obstruction, introduced by Yuri Manin in 1970, explains many failures of the local–global principle for rational points.
• Modular representation theory is a large field built on his foundations: blocks, defect groups, Brauer characters, the main theorems. His height zero conjecture was proved in 2024; the \(k(B)\) conjecture and others from his 1963 list of problems still guide research.
• The classification of finite simple groups followed the path he set in 1954, through centralizers of involutions.
• The Brauer algebra reappeared in knot theory (the Birman–Murakami–Wenzl algebras), statistical mechanics and the representation theory of Lie superalgebras.
• Number theory uses his induction theorem whenever an Artin L-function must be continued, and the Brauer–Siegel theorem in every discussion of class numbers.

Connections

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

Teachers and advisors

• Issai Schur 1926

Students

• Cecil J. Nesbitt
• Donald Passman
• I. Martin Isaacs
• Kenneth A. Fowler
• Louis Solomon
• Paul Fong
• Richard Bruck
• Robert Steinberg

Influenced by

• Carl Ludwig Siegel
• Georg Frobenius
• Erhard Schmidt

Influenced

• John G. Thompson
• Walter Feit

Colleagues and collaborators

• Emil Artin 1941-1947
• Emmy Noether 1927-1933
• Helmut Hasse 1931-1932
• Hermann Weyl 1934-1935
• Konrad Knopp 1925-1933
• Michio Suzuki 1959
• Nathan Jacobson 1934-1935

Family

• Alfred Brauer

Also mentioned in this life

• Augustin-Louis Cauchy
• Richard Dedekind
• Paul Dirac
• Albert Einstein
• Élie Cartan
• Leonhard Euler
• Évariste Galois
• Alexander Grothendieck
• William Rowan Hamilton
• Erich Hecke
• David Hilbert
• Heinz Hopf
• Felix Klein
• Sophus Lie
• Wolfgang Pauli
• Max Planck
• Bernhard Riemann
• Jean-Pierre Serre

Fields

• representation theory 8 lives
• algebra 29 lives
• algebraic number theory 9 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Richard Dagobert Brauer, MacTutor History of Mathematics.
2. J. A. Green, Richard Dagobert Brauer, National Academy of Sciences Biographical Memoirs 75 (1998); adapted from Bull. London Math. Soc. 10 (1978).
3. Brauer, Richard Dagobert, Dictionary of Scientific Biography.
4. Richard Brauer, Wikipedia.
5. National Science Foundation, National Medal of Science: Richard D. Brauer.
6. American Mathematical Society, Richard Dagobert Brauer, 34th President.

Mathematics

1. R. Brauer, Über Systeme hyperkomplexer Zahlen, Math. Z. 30 (1929).
2. R. Brauer, H. Hasse, E. Noether, Beweis eines Hauptsatzes in der Theorie der Algebren, J. reine angew. Math. 167 (1932).
3. R. Brauer and H. Weyl, Spinors in n dimensions, Amer. J. Math. 57 (1935).
4. R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937).
5. R. Brauer and C. Nesbitt, On the modular characters of groups, Ann. of Math. 42 (1941).
6. R. Brauer, On Artin’s L-series with general group characters, Ann. of Math. 48 (1947).
7. R. Brauer, On the zeta-functions of algebraic number fields, Amer. J. Math. 69 (1947).
8. R. Brauer, A characterization of the characters of groups of finite order, Ann. of Math. 57 (1953).
9. R. Brauer and K. A. Fowler, On groups of even order, Ann. of Math. 62 (1955).
10. R. Brauer, Collected Papers, 3 vols., ed. P. Fong and W. J. Wong, MIT Press, 1980.
11. G. Malle, G. Navarro, A. A. Schaeffer Fry, P. H. Tiep, Brauer’s height zero conjecture, Ann. of Math. 200 (2024); arXiv:2209.04736.
12. J.-P. Serre, Linear Representations of Finite Groups, Springer, 1977 (Parts II and III cover induction and modular theory).
13. P. Gille and T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge, 2006.
14. G. Navarro, Characters and Blocks of Finite Groups, Cambridge, 1998.