Paul Gordan

How to use this course

Paul Gordan is remembered for three things: a theorem, a remark and a student. The theorem says that the invariants of a binary form are generated by finitely many of them, and he proved it by an algorithm. The remark, reportedly made about David Hilbert’s non-constructive proof of a far more general theorem, was that it was theology and not mathematics. The student was Emmy Noether. Behind these stories stands a large body of work: the Clebsch–Gordan coefficients of quantum mechanics, Gordan’s lemma on lattice points in cones, the solution of the quintic with Felix Klein, and one of the cleanest proofs that \(e\) and \(\pi\) are transcendental.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04–07 form a short course in classical invariant theory. Lessons 08–10 cover his other mathematics.

Timeline

Breslau, banks and Berlin, 1837–1862

Paul Albert Gordan was born in Breslau on 27 April 1837 to Jewish parents. His father was a merchant and banker. Paul left the gymnasium early for a commercial school and spent several years working in banks, with training periods in Breslau, Geneva and Berlin.

Mathematics drew him out of business. In Berlin he was tutored by Karl Heinrich Schellbach and in 1855 attended Ernst Kummer’s lectures on number theory. He prepared privately for the school-leaving examination, passed it in 1857, and studied mathematics at Breslau, at Königsberg, where the tradition of Carl Gustav Jacob Jacobi was still strong, and at Berlin.

His 1862 dissertation, De linea geodetica, studied geodesics on spheroids and won a prize from the Breslau philosophy faculty. It was a problem in Jacobi’s style, solved by explicit integration. Gordan then went to Göttingen to study Riemann’s theory of Abelian functions. Bernhard Riemann was ill and left for Italy, so the hoped-for contact came to little. In Göttingen Gordan met Johannes Thomae, who encouraged his interest in Riemann’s theory.

Giessen and Clebsch, 1863–1874

In 1863 Alfred Clebsch, newly appointed at Giessen, invited Gordan to join him. Gordan habilitated there that year with a thesis on the transformation of theta functions and became extraordinary professor in 1865. The partnership was one of the most productive of the century.

Gordan brought knowledge of Riemann’s Abelian functions; Clebsch brought algebraic geometry and invariant theory. Their joint book Theorie der Abelschen Functionen (1866) recast Riemann’s theory in algebraic and geometric terms, centred on the Jacobi inversion problem. In 1868 Clebsch introduced Gordan to the invariant theory of binary forms, and within the year Gordan proved the result that made his name (Lesson 05). Arthur Cayley had concluded in 1856 that the binary quintic had infinitely many independent covariants. Gordan’s theorem showed this was wrong, and Cayley acknowledged the error in 1870.

In 1869 Gordan married Sophie Deurer, daughter of a Giessen professor of law. Clebsch moved to Göttingen in 1868 and died of diphtheria in 1872, aged 39. Clebsch and Gordan were the core of a group around the journal Mathematische Annalen, which Clebsch had founded with Carl Neumann in 1868.

Erlangen, 1874

In 1874 Gordan accepted the chair of mathematics at the University of Erlangen in Bavaria. Felix Klein had been professor there since 1872 and had announced his Erlangen Program on arrival. The two worked together intensely on the symmetry groups of the regular solids and their use in solving equations (Lesson 08). Klein left for Munich in 1875. That year Max Noether arrived from Heidelberg, and he became Gordan’s closest colleague and friend for the next 37 years.

Erlangen was a small university, and Gordan spent the rest of his career there. In his 1875 inaugural programme, Über das Formensystem binärer Formen, he gave a revised proof of his finiteness theorem. He admitted openly that his method could not separate the essential generators from superfluous ones. James Joseph Sylvester, who with Cayley was computing lower bounds for the same systems, praised this candour. The English and German schools attacked the same problem from opposite ends: Gordan’s method gave an upper bound on the generating system, the Cayley–Sylvester counting methods a lower bound.

Binary forms and the symbolic method

For the general reader. A binary form is a polynomial in two variables in which every term has the same degree, like \(ax^2+2bxy+cy^2\). When we change coordinates, the coefficients change, but some combinations of them, like \(b^2-ac\), change only by a predictable factor or not at all. These are invariants. Nineteenth-century algebraists wanted a complete list of them for forms of every degree, and Gordan was the master of the computational tools.

Invariants and covariants

A binary form of degree \(n\) is \(f(x,y)=\sum_{i=0}^{n}\binom ni a_i\,x^{n-i}y^{i}\). The group \(\mathrm{SL}_2(\mathbb C)\) acts by linear substitutions of \((x,y)\), and hence on the coefficients \(a_0,\dots,a_n\). A covariant of order \(m\) and degree \(d\) is a polynomial \(C(a;x,y)\), homogeneous of degree \(d\) in the \(a_i\) and of degree \(m\) in \(x,y\), unchanged under this action. An invariant is a covariant of order 0. Covariants form a ring \(\mathcal C_n=\mathbb C[a_0,\dots,a_n,x,y]^{\mathrm{SL}_2}\). The form \(f\) itself is a covariant of degree 1 and order \(n\).

Symbolic notation

Write \(f\) formally as a power of a linear form, \(f=a_x^n=(a_1x+a_2y)^n\), with the rule that \(a_1^{n-i}a_2^{i}\) means the coefficient \(a_i\). Since a product of several coefficients needs several symbols, one introduces equivalent symbols \(f=a_x^n=b_x^n=c_x^n=\cdots\). The bracket \((ab)=a_1b_2-a_2b_1\) is invariant under \(\mathrm{SL}_2\), and \(a_x\) is covariant.

For example, for the quadratic, \((ab)^2\) has each symbol twice. Expanding, \((ab)^2=a_1^2b_2^2-2a_1a_2b_1b_2+a_2^2b_1^2\), which translates to \(a_0a_2-2a_1a_1+a_2a_0=2(a_0a_2-a_1^2)\), minus twice the discriminant.

The quadratic and the cubic

For \(f=ax^2+2bxy+cy^2\), the ring of covariants is \(\mathbb C[f,D]\) with \(D=b^2-ac\). For the cubic \(F=ax^3+3bx^2y+3cxy^2+dy^3\) there are four generators: \(F\); the Hessian \(H=(ac-b^2)x^2+(ad-bc)xy+(bd-c^2)y^2\), a multiple of \(\det\big(\partial^2F\big)\); the Jacobian \(T=\partial(F,H)/\partial(x,y)\) of order 3; and the discriminant \(\Delta=(ad-bc)^2-4(ac-b^2)(bd-c^2)\). They satisfy one relation, the syzygy

The finiteness theorem of 1868

For the general reader. For forms of degree 5, the number of invariants and covariants is infinite, and Cayley believed that no finite list could generate them all. Gordan proved the opposite: for any binary form, finitely many covariants generate the rest by addition and multiplication. He did it by giving an algorithm that produces the list.

The complete systems are: degree 2, two generators; degree 3, four; degree 4, five; degree 5, twenty-three; degree 6, twenty-six. Gordan and his school computed the quintic and sextic systems by hand.

Structure of the proof

Gordan’s tool is the transvectant (Lesson 06). For covariants \(\varphi=\alpha_x^{p}\) and \(\psi=\beta_x^{q}\) written symbolically, the \(k\)-th transvectant is \((\varphi,\psi)_k=(\alpha\beta)^k\alpha_x^{p-k}\beta_x^{q-k}\), for \(0\le k\le\min(p,q)\).

1. Every covariant comes from transvection. Using the first fundamental theorem, every covariant of degree \(d\) is a combination of transvectants \((\varphi,f)_k\) with \(\varphi\) of degree \(d-1\), plus products of covariants of lower degree. So the whole ring is generated by iterated transvectants of \(f\).
2. Systems and relative systems. Suppose \(A\) is a finite generating system for one family of covariants and \(B\) for another. Gordan shows that the transvectants of products \(\big(\prod A_i^{\mu_i},\prod B_j^{\nu_j}\big)_k\) are generated, modulo products of lower terms, by those whose exponent vectors \((\mu,\nu,k)\) are irreducible solutions of a system of linear equations and inequalities with integer coefficients.
3. Finiteness of irreducible solutions. The non-negative integer solutions of such a system form a monoid with finitely many indecomposable elements. This is Gordan’s lemma (Lesson 09). Hence finitely many transvectants suffice at each stage.
4. Induction on order. Gordan builds the system for \(f\) by induction, at each step taking transvectants of the previously constructed systems and using step 3. The induction terminates because the orders of the new covariants decrease.

The proof is constructive but very heavy. In 1890 Hilbert proved finite generation for forms in any number of variables with a few lines (Lesson 07), and Gordan’s method fell out of fashion until computer algebra revived interest in explicit generators.

Transvectants and the Clebsch–Gordan series

For the general reader. When two spinning particles combine, the total spin can take a definite range of values, and the coefficients that describe how the states mix are called Clebsch–Gordan coefficients. Physicists named them after two invariant theorists because the same mathematics appears in the decomposition of products of binary forms, which Clebsch and Gordan worked out decades before quantum mechanics.

Transvectants

Let \(V_n\) be the space of binary forms of degree \(n\), an irreducible \(\mathrm{SL}_2\)-representation of dimension \(n+1\). For \(\varphi\in V_m\), \(\psi\in V_n\) and \(0\le k\le\min(m,n)\), the \(k\)-th transvectant is, up to a non-zero constant,

It is the image of \(\varphi\otimes\psi\) under the operator \(\Omega^k\), where \(\Omega=\partial_{x_1}\partial_{y_2}-\partial_{y_1}\partial_{x_2}\) acts on \(\varphi(x_1,y_1)\psi(x_2,y_2)\), followed by setting both points equal. The first transvectant is the Jacobian; \((\varphi,\varphi)_2\) is twice the Hessian determinant.

Proof sketch. Both sides have dimension \((m+1)(n+1)\), since \(\sum_{k=0}^{n}(m+n-2k+1)=(n+1)(m+1)\) for \(n\le m\). The transvectant maps are \(\mathrm{SL}_2\)-equivariant because \(\Omega\) commutes with the diagonal action, and each is non-zero. Highest weights of \(V_m\otimes V_n\) are \(m+n,m+n-2,\dots\), each with multiplicity one, which forces the decomposition.

Physics dictionary

Spin \(j\) corresponds to \(V_{2j}\), and the monomial \(x^{j+m}y^{j-m}\) to the state \(|j,m\rangle\) up to normalization. The theorem becomes the addition rule for angular momentum: \(j_1\otimes j_2=|j_1-j_2|\oplus\cdots\oplus(j_1+j_2)\). The expansion coefficients of \(|J,M\rangle\) in the basis \(|j_1,m_1\rangle|j_2,m_2\rangle\) are the Clebsch–Gordan coefficients.

Hilbert’s theorem and Gordan’s proof

For the general reader. In 1888 the young David Hilbert proved that invariants are finitely generated in any number of variables, not by finding the generators but by showing that an infinite list was impossible. Gordan, asked to referee the paper, objected to its lack of method. The story that he dismissed it as theology became famous. Eleven years later Gordan published his own short proof of Hilbert’s key lemma, and its method foreshadowed what are now called Gröbner bases.

Hilbert applied it to the ideal generated by all invariants of positive degree. Finitely many invariants generate that ideal, and a degree induction using an averaging operator (Cayley’s \(\Omega\)-process for \(\mathrm{SL}_n\)) shows they generate the invariant ring as an algebra.

The dispute

Gordan refereed Hilbert’s paper for Mathematische Annalen and complained that Hilbert had disdained to present his ideas following formal rules. Hilbert refused to change anything, and Klein, the editor, published it as it stood. The remark “this is not mathematics, this is theology” is attributed to Gordan, but the earliest written record dates from about 25 years later. Gordan in fact admired the young Hilbert and adopted his methods. Later, he is reported to have said that he had convinced himself theology also has its merits.

Gordan’s proof (1899)

Proof. Induction on \(n\). For \(n=1\) take the monomial of least degree. For \(n>1\), write monomials as \(m\,x_n^{e}\) with \(m\) in the first \(n-1\) variables. Let \(M'\) be the set of such \(m\); by induction a finite subset \(\{m_1,\dots,m_r\}\) divides all of them. Let \(E\) be the largest exponent of \(x_n\) among chosen representatives \(m_ix_n^{e_i}\in M\). For each \(e<E\), apply induction to \(\{m: m x_n^{e}\in M\}\). The finitely many monomials collected divide every element of \(M\).

From monomials to ideals. Fix a monomial order (lexicographic, say). For an ideal \(I\), let \(\mathrm{LT}(I)\) be the set of leading monomials of its elements. By the lemma, finitely many \(g_1,\dots,g_s\in I\) have leading monomials dividing every element of \(\mathrm{LT}(I)\). If \(f\in I\), divide: subtract a multiple of some \(g_i\) to cancel the leading term of \(f\), and repeat. Since a monomial order is a well-ordering, the process ends at 0, so \(f\in(g_1,\dots,g_s)\). In modern language, \(\{g_i\}\) is a Gröbner basis, the idea Bruno Buchberger turned into an algorithm in 1965.

The icosahedron and the quintic

For the general reader. The general equation of degree five cannot be solved by radicals. In 1858 Hermite and Kronecker showed it can be solved with elliptic modular functions. In Erlangen in 1874–75, Klein and Gordan found a more geometric route: the rotations of the icosahedron, acting on the sphere, carry exactly the symmetry needed to solve the quintic.

Invariant forms

The rotation group of the icosahedron is \(A_5\), of order 60. Through \(\mathrm{SO}(3)\to\mathrm{PSL}_2(\mathbb C)\) it acts on the Riemann sphere by Möbius transformations and lifts to the binary icosahedral group of order 120 acting on binary forms. Placing two vertices at \(0\) and \(\infty\), the vertices are the roots of

The Hessian of \(f\) is \(121H\), where \(H=-(u^{20}+v^{20})+228(u^{15}v^5-u^5v^{15})-494u^{10}v^{10}\), whose roots are the 20 face centres. The Jacobian of \(f\) and \(H\) is \(20T\), where \(T\) has degree 30 and vanishes at the 30 edge midpoints.

The rational function \(z\mapsto H^3/(1728f^5)\) on the sphere is invariant under \(A_5\) and has degree 60, so it is the quotient map \(\mathbb P^1\to\mathbb P^1/A_5\). Solving the icosahedral equation \(H^3(z)/1728f^5(z)=w\) for \(z\) is the step that replaces radicals. Klein showed that the general quintic reduces, by Tschirnhaus transformations and one square root, to this equation. Gordan’s contribution in 1874–75 was the explicit algebraic machinery connecting the two, and he carried the same programme to the equation of degree 7 with group \(\mathrm{PSL}_2(\mathbb F_7)\) of order 168 (1882, 1885), and later to degree 6 with a group of order 360.

Gordan’s lemma and linear inequalities

For the general reader. To prove his finiteness theorem, Gordan needed a fact about whole-number solutions of linear equations: although there are infinitely many, they are all built by adding up finitely many basic ones. This lemma now underpins toric geometry. In the same circle of ideas he proved one of the first “theorems of the alternative”, an ancestor of linear programming duality.

Proof (cone form). Let \(v_1,\dots,v_r\) be integer vectors generating \(\sigma\) as a cone. Every lattice point \(m\in\sigma\) can be written \(m=\sum\lambda_iv_i\) with \(\lambda_i\ge0\). Split \(\lambda_i=\lfloor\lambda_i\rfloor+\{\lambda_i\}\). Then \(m=\sum\lfloor\lambda_i\rfloor v_i+p\) with \(p=\sum\{\lambda_i\}v_i\) a lattice point in the bounded box \(\{\sum t_iv_i: 0\le t_i<1\}\). The box contains finitely many lattice points, and these together with the \(v_i\) generate the monoid.

The indecomposable elements of the monoid form its Hilbert basis, which is finite and unique for pointed cones. In toric geometry, Gordan’s lemma says the semigroup algebra \(\mathbb C[\sigma^\vee\cap M]\) is finitely generated, so an affine toric variety is an algebraic variety.

Both cannot hold, since then \(0<y^{\mathsf T}Ax=(A^{\mathsf T}y)^{\mathsf T}x=0\). That one must hold is a separation argument: if no \(y\) exists, the convex hull of the rows of \(A\) misses the origin, and a hyperplane separating them gives \(x\). Gyula Farkas’s lemma of 1902 is the best-known relative.

The transcendence of e and π

For the general reader. A transcendental number is not the root of any polynomial with integer coefficients. Charles Hermite proved that \(e\) is transcendental in 1873, and Ferdinand von Lindemann did the same for \(\pi\) in 1882, settling the ancient problem of squaring the circle. Their proofs were long. In 1893 Hilbert, Adolf Hurwitz and Gordan each published simplifications in the same volume of Mathematische Annalen. Gordan’s is close to the version taught today.

Proof

Suppose \(a_0+a_1e+\cdots+a_ne^n=0\) with integers \(a_k\), \(a_0\ne0\). For a polynomial \(g\) of degree \(N\), set \(G=g+g'+\cdots+g^{(N)}\). Integration by parts gives Hermite’s identity

Multiply by \(a_k\) and sum over \(k\). The terms \(a_ke^kG(0)\) sum to 0, so

Now choose a large prime \(p>\max(n,|a_0|)\) and

• At \(x=k\ge1\), \(g\) has a zero of order \(p\), so all derivatives \(g^{(j)}(k)\) are integers divisible by \(p\).
• At \(x=0\), the zero has order \(p-1\). The derivatives of order \(\ge p\) are divisible by \(p\), and \(g^{(p-1)}(0)=\big((-1)^nn!\big)^p\), which is not divisible by \(p\) because \(p>n\).

Hence \(J\) is an integer with \(J\equiv-a_0\big((-1)^nn!\big)^p\not\equiv0\pmod p\), since \(p\nmid a_0\). So \(|J|\ge1\). On the other hand, on \([0,n]\), \(|g(x)|\le n^{(n+1)p}/(p-1)!\), giving \(|J|\le C\cdot c^p/(p-1)!\) for constants \(C,c\) independent of \(p\). This tends to 0 as \(p\to\infty\), a contradiction.

The case of π

Lindemann’s method, simplified in the same way, proves that \(\sum\beta_je^{\alpha_j}\ne0\) whenever the \(\alpha_j\) are distinct algebraic numbers and the \(\beta_j\) non-zero algebraic numbers (the Lindemann–Weierstrass theorem). If \(\pi\) were algebraic, so would be \(i\pi\), and \(e^{i\pi}+1=0\) would contradict the theorem with \(\alpha=i\pi,0\).

The Erlangen years, 1874–1910

For 36 years Gordan and Max Noether ran mathematics at Erlangen. Their joint paper of 1876 studied a claim of Otto Hesse that a form whose Hessian determinant vanishes identically must, after a linear change of variables, involve fewer variables (geometrically, its hypersurface is a cone). Gordan and Noether proved the claim for forms in at most four variables and showed it fails from five variables on. A standard counterexample in five variables is \(x_0x_3^2+x_1x_3x_4+x_2x_4^2\).

His lectures on invariant theory, written up by Georg Kerschensteiner, appeared in two volumes in 1885 and 1887. Kerschensteiner later became one of Germany’s best-known educational reformers. A third volume was announced but never appeared. Gordan published over eighty papers, mostly in Mathematische Annalen. His style was computational: he derived results by long chains of explicit formulas and rarely explained the ideas behind them.

Around 1900, with G. Alexejeff, he tried to apply invariant theory to chemical valence, treating molecules as systems of binary forms. Eduard Study criticized the project sharply, chemists showed little interest, and it was dropped.

Gordan was elected to the Bavarian Academy of Sciences in 1886, received an honorary doctorate from Dublin, and was an honorary member of the London Mathematical Society for more than three decades. Contemporaries called him the king of invariant theory.

Noether, death and legacy, 1907–1912

Emmy Noether, Max Noether’s daughter, enrolled at Erlangen in 1904 and wrote her dissertation under Gordan, Über die Bildung des Formensystems der ternären biquadratischen Form (1907). It applied his symbolic methods to ternary quartics and computed a system of 331 covariants. She later disowned the work. The Mathematics Genealogy Project also lists Sigmund Gundelfinger (1867), Harry Tyler (1889), Edward Drake Roe (1898) and Wilhelm Hauser (1907) as his doctoral students, though some biographies treat Noether as his only student.

Gordan retired in 1910 and died in Erlangen on 21 December 1912, aged 75. Max Noether wrote his obituary for Mathematische Annalen (1914), with a complete bibliography. In 1911 Ernst Fischer took the Erlangen chair; he steered Emmy Noether toward Hilbert’s abstract methods.

Legacy

• Invariant theory: the first finiteness theorem and the constructive tradition, revived by computational invariant theory and Gröbner bases.
• Physics: the Clebsch–Gordan series and coefficients, standard in quantum angular momentum and the representation theory of Lie groups.
• Combinatorics and geometry: Gordan’s lemma, the foundation of affine toric varieties and of Hilbert bases in integer programming; his theorem of the alternative in optimization.
• Algebra and Galois theory: with Klein, the geometric solution of the quintic.
• Number theory: the streamlined transcendence proofs for \(e\) and \(\pi\).
• Teaching: through Emmy Noether, the bridge from nineteenth-century computation to twentieth-century abstract algebra.

Connections

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

Teachers and advisors

• Carl Gustav Jacob Jacobi

Students

• Emmy Noether 1907
• Edward Drake Roe 1898
• Harry Tyler 1889
• Sigmund Gundelfinger 1867
• Wilhelm Hauser 1907

Influenced by

• Arthur Cayley
• Bernhard Riemann
• Carl Gustav Jacob Jacobi
• Ernst Kummer 1855

Colleagues and collaborators

• David Hilbert 1888-1912
• Felix Klein 1874-1875
• Max Noether 1875-1912
• Alfred Clebsch 1863-1872
• Ernst Fischer
• Georg Kerschensteiner 1885-1887

Rivals

• Arthur Cayley
• David Hilbert
• James Joseph Sylvester

Also mentioned in this life

• Évariste Galois
• Charles Hermite
• Adolf Hurwitz
• Leopold Kronecker
• Sophus Lie
• August Ferdinand Möbius
• Karl Weierstrass

Fields

• invariant theory 7 lives
• algebra 29 lives
• algebraic geometry 17 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Paul Gordan, MacTutor History of Mathematics.
2. Paul Gordan, Wikipedia.
3. N. Stuloff, Gordan, Paul, Neue Deutsche Biographie 6 (1964).
4. Gordan, Paul Albert, Dictionary of Scientific Biography.
5. Obituary of Paul Gordan, Proceedings of the London Mathematical Society.
6. Paul Albert Gordan, Mathematics Genealogy Project.
7. M. Noether, Paul Gordan, Mathematische Annalen 75 (1914).

Mathematics

1. A. Clebsch and P. Gordan, Theorie der Abelschen Functionen, Teubner, 1866.
2. P. Gordan, Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist, J. reine angew. Math. 69 (1868).
3. P. Gordan, Vorlesungen über Invariantentheorie, ed. G. Kerschensteiner, 2 vols., Teubner, 1885–87.
4. P. Gordan and M. Noether, Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet, Math. Ann. 10 (1876). See also A. Garbagnati and F. Repetto, arXiv:0802.0959.
5. P. Gordan, Transcendenz von \(e\) und \(\pi\), Math. Ann. 43 (1893).
6. F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Teubner, 1884.
7. B. Sturmfels, Algorithms in Invariant Theory, Springer, 2nd ed., 2008.
8. P. J. Olver, Classical Invariant Theory, Cambridge University Press, 1999.
9. Gordan’s lemma and Clebsch–Gordan coefficients, Wikipedia.