Emmy Noether

How to use this course

Emmy Noether is known to physicists for one theorem and to mathematicians for a whole way of thinking. Physicists learn that every continuous symmetry of an action gives a conservation law. Algebraists learn rings, ideals and modules in the language she created, often without knowing it was hers. This course presents both, alongside the story of a career spent largely without pay, title or security.

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 the chronology of her work: invariant theory (1907–1916), variational problems (1918), commutative algebra (1921–1927), and noncommutative algebra (1927–1933).

Timeline

Erlangen, 1882–1907

Amalie Emmy Noether was born in Erlangen on 23 March 1882 into a prosperous Jewish family. Her father, Max Noether, was a professor at the university and a leading algebraic geometer. Her younger brother Fritz also became a mathematician. As a girl she was ordinary in the eyes of her teachers: short-sighted, friendly, fond of dancing. She followed the path open to educated girls and in 1900 qualified to teach English and French.

Instead of teaching, she began sitting in on lectures at Erlangen. Women could not enrol in Bavarian universities, and auditors needed each professor’s permission. In 1903 she was one of two women among 986 students. That winter she went to Göttingen and heard Hilbert, Klein and Minkowski. In 1904 Bavaria opened its universities to women, and she enrolled at Erlangen, the only woman in mathematics in her year.

The thesis

Her advisor was Paul Gordan, a friend of her father and the master of the computational school of invariant theory. Her 1907 dissertation, Über die Bildung des Formensystems der ternären biquadratischen Form, computed a complete system of 331 covariants of a ternary quartic form, by hand, in Gordan’s symbolic style. It earned a summa cum laude. Years later she dismissed the work with a single rude word in German. Lesson 04 explains why her view of invariant theory changed so completely.

The unpaid years, 1908–1915

With a doctorate but no possibility of a university post, Noether stayed at the Erlangen Mathematical Institute for seven years without salary. She supervised doctoral students whose degrees were officially under her father, substituted for him when he was ill, and kept publishing. She was elected to the Circolo Matematico di Palermo in 1908 and to the German Mathematical Society in 1909.

The decisive influence of these years was Ernst Fischer, who succeeded Gordan in 1911. Through long conversations, often continued on postcards, Fischer turned her from Gordan’s explicit computations toward Hilbert’s abstract, existential methods. Her papers of this period include work on the field of rational invariants and the question now called Noether’s problem: when is the field of invariants of a finite group acting on \(k(x_1,\dots,x_n)\) itself a purely transcendental extension? The question is a route to the inverse Galois problem, and it stayed open until counterexamples were found in 1969.

Göttingen and the habilitation, 1915–1919

In 1915 David Hilbert and Felix Klein invited her to Göttingen. Einstein had just completed general relativity, and Hilbert and Klein were struggling with the role of energy conservation in the new theory. They wanted an expert in invariant theory.

To lecture in a German university she needed the habilitation. The mathematicians and scientists supported her; the historians and philologists of the faculty objected that soldiers returning from the front should not be asked to learn at the feet of a woman. Hilbert’s retort that a faculty senate is not a bathhouse is often quoted, though its exact wording is uncertain. The application failed. For four years her courses were announced under Hilbert’s name, with her listed as his assistant.

In this period she wrote the paper that physics still uses daily. On 26 July 1918 Klein presented Invariante Variationsprobleme to the Royal Society of Sciences in Göttingen, of which she could not be a member. Einstein wrote to Hilbert praising its penetrating mathematical thinking. After the war, the new Weimar Republic’s constitution proclaimed the equality of men and women, and in 1919 her habilitation was finally approved. She was 37.

Invariant theory and finiteness

For the general reader. A polynomial is invariant under a group if the group’s symmetries leave it unchanged. Nineteenth-century mathematicians tried to list all invariants explicitly. Noether began in that tradition and ended by proving finiteness theorems so clean that the long lists became unnecessary.

From Gordan to Hilbert

Let a group \(G\) act linearly on a vector space \(V\) over a field \(k\), and hence on the polynomial ring \(k[V]=k[x_1,\dots,x_n]\). The invariant ring is \(k[V]^G=\{f : g\cdot f=f \ \forall g\in G\}\). Gordan proved in 1868, by explicit construction, that invariants of binary forms under \(\mathrm{SL}_2\) are finitely generated. In 1890 Hilbert proved finite generation in far greater generality using his basis theorem, without constructing a single generator. Gordan’s reaction, that this was theology rather than mathematics, is famous. Noether’s thesis was the last great work of the computational school; her later work belongs entirely to Hilbert’s.

Proof sketch

The Reynolds operator \(R(f)=\frac{1}{|G|}\sum_{g\in G} g\cdot f\) projects \(k[V]\) onto \(k[V]^G\), so the invariants are spanned by the averages \(R(x^\alpha)\) of monomials. Introduce new variables \(t_1,\dots,t_n\), set \(\ell=\sum t_i x_i\), and consider the power sums

The \(p_m\) are power sums of the \(|G|\) quantities \(g\cdot\ell\). By Newton’s identities, every \(p_m\) is a polynomial (with rational coefficients) in \(p_1,\dots,p_{|G|}\). Comparing coefficients of \(t^\alpha\) shows that every \(R(x^\alpha)\) is a polynomial in the \(R(x^\beta)\) with \(|\beta|\le|G|\). Characteristic 0 is used twice: to divide by \(|G|\) and by multinomial coefficients, and in Newton’s identities.

Every characteristic (1926)

For each variable \(x_i\), the polynomial \(\prod_{g\in G}(T-g\cdot x_i)\) has invariant coefficients. Let \(B\) be the subalgebra generated by all these coefficients: it is finitely generated, hence Noetherian by Hilbert’s basis theorem, and \(k[V]\) is a finite \(B\)-module because each \(x_i\) is integral over \(B\). Then \(k[V]^G\) is a \(B\)-submodule of a finite module over a Noetherian ring, hence itself finite over \(B\), hence a finitely generated algebra. The argument is three lines once the right concepts exist, and Noether supplied the concepts (Lesson 07).

The first theorem: symmetry and conservation

For the general reader. Why is energy conserved? Noether’s answer: because the laws of physics are the same today as tomorrow. Momentum is conserved because they are the same here as there, and angular momentum because they do not prefer a direction. Each continuous symmetry gives a conserved quantity, and conversely.

Mechanics

Let \(L(q,\dot q,t)\) be a Lagrangian with \(q\in\mathbb R^N\). Suppose the infinitesimal transformation \(q\mapsto q+\varepsilon\,\xi(q,t)\) changes \(L\) only by a total derivative: \(\delta L=\varepsilon\,\frac{dF}{dt}\). Using the Euler–Lagrange equations \(\frac{d}{dt}\frac{\partial L}{\partial\dot q}=\frac{\partial L}{\partial q}\),

Time translation, treated the same way, gives the energy \(E=\sum_i p_i\dot q_i-L\) whenever \(\partial L/\partial t=0\).

Field theory

Let \(S[\phi]=\int L(x,\phi,\partial\phi)\,d^nx\). Noether considered a group \(G_\rho\) of transformations depending on \(\rho\) parameters, a Lie group in modern terms.

For an internal symmetry \(\delta\phi=\varepsilon\,\Delta\phi\) with \(\delta L=\varepsilon\,\partial_\mu K^\mu\), the current is

Spacetime translations give the canonical energy–momentum tensor \(T^\mu{}_\nu=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\nu\phi-\delta^\mu_\nu L\), with \(\partial_\mu T^\mu{}_\nu=0\). Noether’s paper also allowed generalized symmetries depending on derivatives of \(\phi\), which are now essential in the theory of integrable systems.

The second theorem and energy in general relativity

For the general reader. Hilbert had noticed that energy conservation in general relativity behaves strangely: it seems to hold automatically, without using the field equations. Noether explained why. When the symmetry depends on arbitrary functions, as in general relativity, it produces identities rather than genuine conservation laws.

Two examples

Electromagnetism. \(L=-\tfrac14F_{\mu\nu}F^{\mu\nu}\) with \(F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu\) is invariant under \(A_\mu\mapsto A_\mu+\partial_\mu\lambda(x)\), one arbitrary function. The Euler–Lagrange expressions are \(E^\nu=\partial_\mu F^{\mu\nu}\), and the Noether identity is \(\partial_\nu E^\nu\equiv0\). Coupling to a source current then forces \(\partial_\nu J^\nu=0\).

General relativity. The Einstein–Hilbert action \(\int R\sqrt{-g}\,d^4x\) is invariant under diffeomorphisms, four arbitrary functions. The Euler–Lagrange expressions are the components of the Einstein tensor \(G^{\mu\nu}\), and the four Noether identities are the contracted Bianchi identities

“Improper” conservation laws

When the finite-dimensional group of Lesson 05 sits inside a group depending on arbitrary functions, Noether showed that the resulting currents have the form \(j^\mu=\partial_\nu U^{\mu\nu}+(\text{terms vanishing on solutions})\) with \(U^{\mu\nu}\) antisymmetric. Such a current is conserved identically, and its charge reduces to a surface integral at infinity. This is the precise sense in which local energy conservation in general relativity differs from that in special relativity, the point that had troubled Hilbert, Klein and Einstein. The same structure underlies Gauss’s law in gauge theories and the ADM mass.

Ascending chains and primary decomposition

For the general reader. Every whole number factors into primes. In 1921 Noether showed that a weaker kind of factorization holds in a vast class of rings, and that it rests on a single finiteness condition: you cannot keep enlarging an ideal forever. Rings with this property now bear her name.

For (1)⇒(3): if \(I\) were not finitely generated, choose \(a_1,a_2,\dots\in I\) with \(a_{k+1}\notin(a_1,\dots,a_k)\), producing a strictly increasing chain. For (3)⇒(1): the union of a chain is an ideal, generated by finitely many elements that all lie in some \(I_k\). Rings satisfying these conditions are Noetherian. Hilbert’s basis theorem says that \(R\) Noetherian implies \(R[x]\) Noetherian.

Primary decomposition

An ideal \(\mathfrak q\ne R\) is primary if \(ab\in\mathfrak q\) and \(a\notin\mathfrak q\) imply \(b^n\in\mathfrak q\) for some \(n\). Its radical \(\sqrt{\mathfrak q}=\mathfrak p\) is then prime. Emanuel Lasker, the world chess champion, had proved decomposition for polynomial rings in 1905 by heavy computation. Noether derived it from ACC alone.

Proof idea. Call \(I\) irreducible if \(I=J\cap K\) forces \(I=J\) or \(I=K\). If some ideal were not a finite intersection of irreducibles, the set of such ideals would have a maximal element \(I\) (condition 2); \(I\) is reducible, so \(I=J\cap K\) with \(J,K\) strictly larger, each a finite intersection of irreducibles, a contradiction. This argument is Noetherian induction. Then one shows, again with ACC applied to the chain \((I:b)\subseteq(I:b^2)\subseteq\cdots\), that irreducible ideals are primary.

Dedekind domains and normalization

For the general reader. In some number systems, numbers do not factor uniquely into primes. Dedekind restored uniqueness by factoring ideals instead. In 1927 Noether found exactly which axioms make his theory work, turning a collection of special results into a short list of properties.

Such rings are called Dedekind domains. Rings of integers of number fields and coordinate rings of smooth affine curves satisfy condition 2, which is easy to check, so they enjoy condition 1.

Geometrically, every affine variety of dimension \(d\) maps finitely onto affine \(d\)-space. Normalization is the standard route to Hilbert’s Nullstellensatz.

Modules, representations, homology

For the general reader. Noether’s habit was to replace calculations with structures. She saw that a representation of a group is just a module over a ring, and that the numbers topologists counted were shadows of groups. Both ideas are now so standard that they seem obvious.

Representations as modules (1929)

A representation of a finite group \(G\) on a \(k\)-vector space \(V\) is the same thing as a module over the group algebra \(k[G]\). Subrepresentations are submodules, irreducibles are simple modules, and intertwiners are module homomorphisms. In this language the classical theorems of Frobenius, Burnside and Schur become statements about rings:

Her 1927 work also stated the isomorphism theorems in the general form now taught for groups, rings and modules alike.

Homology groups

In 1925 and 1926, through her contact with Pavel Alexandrov and Heinz Hopf, Noether insisted that the Betti numbers and torsion coefficients of a space should be read as invariants of groups: cycles modulo boundaries. Hopf, Vietoris and Mayer soon wrote topology in this form, and Hopf credited her. Algebraic topology has been the study of homology groups ever since.

Brauer–Hasse–Noether

For the general reader. Hamilton’s quaternions are a number system where multiplication is not commutative. Noether, with Richard Brauer and Helmut Hasse, classified all such division algebras over number fields by a local-to-global principle: an algebra is trivial exactly when it is trivial at every prime. The result became a cornerstone of class field theory.

Central simple algebras

A central simple algebra over a field \(K\) is a finite-dimensional \(K\)-algebra with centre \(K\) and no two-sided ideals besides 0 and itself. By Wedderburn, \(A\cong M_n(D)\) for a division algebra \(D\). Classes under \(A\sim M_n(A)\) form the Brauer group \(\mathrm{Br}(K)\) under \(\otimes_K\). Noether developed crossed products: for a Galois extension \(L/K\) with group \(G\), algebras split by \(L\) are described by factor systems, so that \(\mathrm{Br}(L/K)\cong H^2(G,L^\times)\) in today’s Galois cohomology, which her work anticipated.

Together with Hasse’s local invariants this gives the exact sequence

where \(\mathrm{Br}(K_v)\cong\mathbb Q/\mathbb Z\) for finite \(v\), \(\tfrac12\mathbb Z/\mathbb Z\) for real \(v\), and 0 for complex \(v\).

The Noether school, 1919–1933

As a Privatdozentin Noether had the right to teach but no salary. In 1922 she received the title of extraordinary professor, still unpaid; from 1923 a teaching assignment in algebra brought a modest income. She never held a regular chair.

She built a school anyway. Students and visitors, known affectionately as the Noether boys, though they included women such as Grete Hermann and Olga Taussky, followed her lectures and her long walks through Göttingen. Her lectures were fast, improvised and hard to follow, and they were full of ideas she gave away freely. Her doctoral students included Grete Hermann, Max Deuring, Hans Fitting, Jakob Levitzki, Ernst Witt and Chiungtze Tsen. Bartel van der Waerden, who came from Amsterdam to work with her, wrote Moderne Algebra (1930–31) from the lectures of Noether and Emil Artin; it taught the new algebra to the world. Colleagues joked that anything new in algebra could already be found somewhere in Noether.

She spent the winter of 1928–29 in Moscow, teaching abstract algebra and algebraic geometry and working with Pavel Alexandrov, who became a close friend. Her politics were pacifist and on the left, which later counted against her. In 1932 she shared the Ackermann–Teubner Memorial Prize with Artin and gave a plenary address at the International Congress of Mathematicians in Zürich, the high point of her public recognition.

Exile, death and legacy, 1933–1935

In April 1933 the new Nazi government’s civil-service law removed Jewish academics from their posts, and Noether lost the right to teach at Göttingen. Her brother Fritz was dismissed from Breslau and emigrated to the Soviet Union. She continued to teach students privately in her apartment for a time, reportedly including one who arrived in a brownshirt uniform.

With help from the Rockefeller Foundation and the Emergency Committee in Aid of Displaced Foreign Scholars, she went to Bryn Mawr College in Pennsylvania in the autumn of 1933, where Anna Pell Wheeler welcomed her. She taught a small group of women students and from 1934 lectured weekly at the Institute for Advanced Study in Princeton. In April 1935 she underwent surgery for a tumour. She seemed to recover, then died suddenly on 14 April 1935, aged 53. Her ashes rest beneath the walkway around the cloisters of the old library at Bryn Mawr.

Hermann Weyl gave her memorial address at Bryn Mawr. In a letter to the New York Times in May 1935, Einstein called her the most significant creative mathematical genius produced since the higher education of women began.

Legacy

• Physics: the link between symmetries and conservation laws organizes classical mechanics, quantum field theory and the Standard Model.
• Algebra: Noetherian rings and modules, primary decomposition, normalization and the axiomatic method are the foundation of commutative algebra and algebraic geometry, from Krull to Grothendieck.
• Topology and number theory: homology groups, and the cohomological view of the Brauer group.
• Honours: the Association for Women in Mathematics has held an annual Noether Lecture since 1980; the International Mathematical Union holds an Emmy Noether Lecture at each Congress; the German Research Foundation’s programme for young researchers bears her name.

Connections

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

Teachers and advisors

• Paul Gordan 1907

Students

• Bartel van der Waerden 1924-1928
• Olga Taussky-Todd
• Ernst Witt
• Grete Hermann 1926
• Max Deuring

Influenced by

• Richard Dedekind
• Ernst Fischer 1911-1915

Influenced

• Alexander Grothendieck
• André Weil 1927
• Bartel van der Waerden 1924-1933
• Heinz Hopf
• Oscar Zariski 1935-1936
• Oswald Teichmüller

Colleagues and collaborators

• Albert Einstein 1918-1935
• Carl Ludwig Siegel 1930
• David Hilbert 1915-1933
• Emil Artin
• Felix Klein 1915-1925
• Helmut Hasse 1927-1933
• Hermann Weyl
• Olga Taussky-Todd 1931-1935
• Richard Brauer 1927-1933
• Richard Courant 1920-1933

Friends

• Pavel Alexandrov 1923-1935

Family

• Max Noether

Also mentioned in this life

• Leonhard Euler
• Georg Frobenius
• Évariste Galois
• Carl Friedrich Gauss
• William Rowan Hamilton
• Joseph-Louis Lagrange
• Sophus Lie
• Hermann Minkowski
• Isaac Newton
• Osborne Reynolds
• Issai Schur

Fields

• abstract algebra 4 lives
• invariant theory 7 lives
• mathematical physics 45 lives

On CoursesHub

• Noether & the Symmetry Principle · History of Mathematics & Physics

Sources and further reading

Biography

1. Emmy Noether, Wikipedia (with extensive references).
2. Jewish Women’s Archive, Emmy Noether.
3. A. Einstein, Letter to the editor, New York Times, May 1935.
4. N. Byers, The life and times of Emmy Noether, arXiv:hep-th/9411110.
5. A. Dick, Emmy Noether 1882–1935, Birkhäuser, 1981.
6. H. Weyl, Emmy Noether, Scripta Mathematica 3 (1935).

Mathematics

1. E. Noether, Invariante Variationsprobleme, Nachr. Ges. Wiss. Göttingen (1918). English translation by M. A. Tavel.
2. E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77 (1916).
3. E. Noether, Idealtheorie in Ringbereichen, Math. Ann. 83 (1921).
4. E. Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, Math. Ann. 96 (1927).
5. E. Noether, Hyperkomplexe Größen und Darstellungstheorie, Math. Z. 30 (1929).
6. R. Brauer, H. Hasse, E. Noether, Beweis eines Hauptsatzes in der Theorie der Algebren, J. reine angew. Math. 167 (1932).
7. Y. Kosmann-Schwarzbach, The Noether Theorems, Springer, 2011.
8. The Noether theorems in context, arXiv:2004.09254.
9. Colloquium: A century of Noether’s theorem, arXiv:1902.01989.
10. B. L. van der Waerden, Moderne Algebra, Springer, 1930–31.