The Assault on the Quintic Niels Henrik Abel

• Niels Henrik Abel

• Niels Henrick Abel was born in 1802 to a Lutheran Minister and the daughter of a shipping merchant. • His father came from a long line of pastors; his mother was a beautiful woman with a passion for the finer things. Niels Henrick Abel

• Poverty and alcoholism haunted the family; at the same time, famine haunted Norway. • Abel attended Cathedral school in Oslo and along with many other students was beaten by the mathematics teacher. Abel

• Being with his friends and going to the theater became Abel’s salvation. • Bernt Michael Holmboe replaced the sadistic math teacher and recognized Abel’s talent. They remained friends throughout Abel’s life. Abel

• During these high school years he thought he’d solved the general quintic. He submitted the proof and was asked to better explain it. In trying to do so, he found his mistake. Abel

• His father died in 1820, making even worse the poverty that was to remain with him all his life. Somehow, in spite of this poverty, he entered the university in 1821. • During this time, two professors supported him, and even helped him obtain funds for travel to Denmark to meet other mathematicians. Christine (Crelly) Kemp

• During this trip he met his future fiancée Crelly Kemp; they were engaged in 1824. • They met at a ball; he asked her to dance, and she said yes. Then, after several fruitless attempts, they stopped, and burst out laughing. Neither knew how to dance. • Abel wrote of Crelly, “She is not beautiful, has red hair and freckles, but she is a wonderful girl.” Abel

• He wrote a proof that the quintic was not solvable by radicals in 1823. Understanding its importance, he wrote a very brief version of the proof and used it as a sort of “business card” to send to mathematicians. • The one he sent to Gauss was never opened, and was found among Gauss’s papers after his death. Abel

• Through the efforts of his teachers in Norway, Abel received a travel grant in 1825. There was, unfortunately, no guarantee of support upon his return. • He was meant to go to Paris, to meet Cauchy, Legendre and others, but because his friends were going to Berlin he went there instead. • He met August Crelle, who founded a journal still in publication today. The first volume in 1826 included seven papers by Abel. Crelle also began working to secure Abel a teaching position in Berlin. • During this period of time a position came open at the university in Oslo, and Abel applied for it. He lost out to his friend Holmboe, and although it was a big disappointment to Abel they remained good friends. • In general, Abel spent a happy winter in Berlin in 1825, with friends, parties and writing of mathematical papers. Abel

• Abel made his trip to Paris in the late summer of 1826. He met Legendre and Cauchy, but their meetings were quite formal. They paid little attention to him, being absorbed in their own work and interests. • Abel continued to work by himself. He submitted a work on transcendental functions to the French Academy of Sciences that was mislaid by Cauchy and ignored by Legendre. • Eventually his money ran out and his health problems began. What turned out to be tuberculosis (although Abel never did admit it) was in its beginning stages. He returned to Berlin, where his health worsened. Abel

• In May 1927, he went back to Oslo. He was given a very small stipend and eventually a substitute teaching job. He continued his work on elliptic integrals. • During this period, his fame was beginning to spread in Europe. He spent the summer with Crelly, which was again a relatively happy respite. • He returned to Oslo in fall 1828. His income and health both declined. • Finally, on April 6, 1829 he died, with Crelly at his bedside. • Two days later, a letter from Crelle arrived saying he had been offered a position in Berlin. Abel’s Proof

• Proved that the general quintic equation could not be solved by radicals. • Outline of proof: First, Abel showed that if a quintic polynomial had a solution by radicals, it had to be in the form ଵൗ ଶൗ ଷൗ ସൗ ହ ଶ ହ ଷ ହ ସ ହ Where the p’s are finite sums of radicals and polynomials, and the R’s are irrational functions of the coefficients of the quintic. (Notice the similarity to the form suggested by Euler: ೙ ೙ ଶ ೙ ଷ ) Abel’s Proof

• Second, assuming there was a solution in this form, he showed (following Vandermonde and Lagrange) that this expression must be equal to a rational function of all the roots of the equation.

• Ugly rational function of 5 roots = ugly irrational function of 5 coefficients. Abel’s Proof

• Third, using a result of Cauchy’s, he proved that the two expressions could not be equal, because permuting the roots leads to inconsistent numbers of possible values on the left and right sides. • This contradiction establishes that no such formula (from the right side) can exist. • This is a gross oversimplification and leaves out many good bits, but gives you the flavor of the proof. Abel’s Mathematics

• Abel also worked in the field of elliptic integrals. • elliptic integrals originally arose from attempts to find by calculus the arc length of an ellipse. Abel’s Mathematics

• Today, we think of them as any function that can be expressed in the form: ௫

௖ Where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant. Abel’s Mathematics

• From the Encyclopedia Britannica: • [Abel’s paper] dealt with the sum of integrals of a given algebraic function. Abel's theorem states that any such sum can be expressed as a fixed number p of these integrals, with integration arguments that are algebraic functions of the original arguments. …Abel's theorem is a vast generalisation of Euler's relation for elliptic integrals. • This is the paper that Cauchy lost. Abel’s Mathematics

• In 1828 Abel was shown a paper by Jacobi on transformations of elliptic integrals. Abel showed that Jacobi’s work followed from his own (from the lost manuscript) and published a second work in this area. • He had been working again on the algebraic solution of equations, with the aim of solving the problem of which equations were soluble by radicals (the problem which Galois solved a few years later). He put this to one side to compete with Jacobi in the theory of elliptic functions, quickly writing several papers on the topic. Abel’s Mathematics

• 1n 1830 both Abel and Jacobi were awarded the Grand Prix for their work. • Of their work, Legendre said, “Through these works you two will be placed in the class of the foremost analysts of our times.” • Curiously enough, the original lost manuscript was found by Cauchy in 1830. It was printed in 1841, then vanished again to surface in 1952 in Florence. Abel’s Mathematics

• After Abel's death unpublished work on the algebraic solution of equations was found. In fact in a letter Abel had written to Crelle on 18 October 1828 he gave the theorem: • If every three roots of an irreducible equation of prime degree are related to one another in such a way that one of them may be expressed rationally in terms of the other two, then the equation is soluble in radicals. Abel’s Mathematics

• This result is essentially identical to one given by Galois in his famous memoir of 1830. Evariste Galois Evariste Galois

• Born 25 Oct 1811 in Bourg La Reine, near Paris. • Parents were well educated and taught Evariste Greek, Latin and religion at home. • During the last part of Napoleon’s reign. Evariste Galois

• Entered Lycée of Louis‐ le‐Grand in 1823. Generally a good students, but had to repeat a year because his “rhetoric” was not up to standard. • In 1827, enrolled in his first mathematics class. Galois

• He quickly lost interest in everything else. A director of the school wrote of this, • It is the passion for mathematics which dominates him, I think it would be best for him if his parents would allow him to study nothing but this, he is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments. Galois

• Wanted to enter the École Polytechnique, partly for academic reasons and partly for political reasons. He (and his parents) were ardent Republicans and the Polytechnique had a strong student movement that drew Evariste’s interest. • Didn’t pass the entrance exam (likely because he concentrated more on mathematics than rhetoric). Galois

• Continued to work on mathematics, having a paper published on continued fractions in April 1829. • In July 1929 his father hanged himself, probably as a result of a scandal involving his forged name on some libelous letters. • A few weeks later, Evariste sat for his second entrance exam for the École Polytechnique, and again failed. Galois

• He resigned himself to entering the École Normale, and sat for his baccalaureate exams. He passed, doing well in mathematics, although his literature examiner said, • This is the only student who has answered me poorly, he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly, for, after his examination, I believed him to have but little intelligence. Galois

• He sent work to Cauchy on the solvability of equations, and then learned of Abel’s posthumously published work. At Cauchy’s suggestion, he submitted a work on the conditions under which equations were solvable by radicals in 1830. He also worked some on elliptic integrals, having seen Abel’s and Jacobi’s work, but learned that they received the prize without his work being considered (Fourier had lost it, then died). Galois

• At about this time, Galois became involved in the revolution, for which he was expelled from the École Normale. He joined a republican arm of the militia, which was soon disbanded as a threat to the throne. • Late in 1830, he was arrested for making threats against the king, and held in prison for the length of his trial, but was acquitted. Galois

• On Bastille Day of 1831 he was arrested for wearing the uniform of the Artillery of the National Guard, which was illegal. He was also carrying a loaded rifle, several pistols and a dagger. • He was sent back to prison. He tried to commit suicide but was prevented from doing so by the other prisoners. Galois

• He also received word that his manuscript was rejected, although he was encouraged to write a more clear and complete account of his work. • He admitted to others while drunk in prison, “Do you know what I lack my friend? I confide it only to you: it is someone whom I can love and love only in spirit. I have lost my father and no one has ever replaced him, do you hear me...? Galois

• He was transferred to a hotel because of a cholera outbreak, and there he fell in love with the resident physician’s daughter, Stephanie‐ Felice du Motel. He wrote “Stephanie” several times in the margins of one of his mathematical works. Galois

• For reasons that are not entirely clear (though almost certainly connected to Stephanie), Galois ended up in a duel with Perscheux d'Herbinville on May 30, 1832. • Some sources say only one shot was fired. • Galois was wounded and abandoned by both his opponent and his own seconds. He was found by peasants and taken to a hospital, where he died on May 31. The Night Before

• The legend was that Galois spent the night before his duel feverishly writing down everything he knew about group theory and solution by radicals. • A marginal note has been partly to blame for this idea. The Night Before

• The note reads, There is something to complete in this demonstration. I do not have the time. (Author's note). • The legend appears to have been an exaggeration. Galois

• It was Galois’ wish that his mathematical work be sent to Gauss and Jacobi for their opinions. • His brother and his friend copied his papers and send them to Gauss, Jacobi and several others. There is no record of these two mathematicians offering an opinion (Gauss didn’t think much of solutions by radicals – not an important question for him). Galois

• Louiville had this to say of one of his results, however: • [There is a solution]...as correct as it is deep of this lovely problem: Given an irreducible equation of prime degree, decide whether or not it is soluble by radicals. • Louiville published it in his journal in 1846. Galois

• Today, we still study Galois Theory in abstract algebra (in particular, in Math 372 at BYU). • And we talk of Abelian groups. Ruffini Ruffini

• Paolo Ruffini was born 22 September, 1765 in Valentano, Italy. • His father was a medical doctor. • Paolo studied mathematics, medicine, philosophy, and literature at the University of Modena. Ruffini

• He graduated with degrees in philosophy, medicine, surgery, and mathematics. • He lost a position teaching mathematics after Napoleon’s taking over Modena. Ruffini would not swear an oath of allegiance to the republic. He took this (and most things) in stride, turning instead to his medical practice. Ruffini

• In 1799 he published a book entitled, General Theory Of Equations In Which It Is Shown That The Algebraic Solution Of The General Equation Of Degree Greater Than Four Is Impossible. Ruffini

• The introduction to the book reads, The algebraic solution of general equations of degree greater than four is always impossible. Behold a very important theorem which I believe I am able to assert (if I do not err): to present the proof of it is the main reason for publishing this volume. The immortal Lagrange, with his sublime reflections, has provided the basis of my proof. Ruffini

• Building on the work of Lagrange, Ruffini introduced several group‐theoretical ideas and proved several theorems on his way to the main theorem. • Except for one gap, Ruffini’s proof did establish that the general quintic could not be solved by radicals. – The gap was in a proof that all algebraic functions could be expressed in terms of rational functions of the roots of an equation, which Abel did prove. Ruffini

• Ruffini sent his book three different times to Lagrange, whose work he clearly acknowledged in the book. He got no reply. • In fact, his book seemed to be largely ignored by the major mathematicians of the time. • Ruffini reacted by writing a second proof which he hoped would be more accepted: In the present memoir, I shall try to prove the same proposition [insolubility of the quintic] with, I hope, less abstruse reasoning and with complete rigour. Ruffini

• He wrote more proofs in 1808 and 1813. • He asked the Institute in Paris to pass judgment on the correctness of his proof. Lagrange, Legendre, and Lacroix were asked to pass judgment. Lagrange “found little in it worthy of attention.” • Strangely enough, it was Cauchy who finally acknowledged Ruffini’s work. Ruffini

• A year before Ruffini’s death, Cauchy wrote to him, “... your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgment, proves completely the impossibility of solving algebraically equations of higher than the fourth degree.” Ruffini

• In 1817, there was a typhus epidemic and Ruffini, who continued to treat his patients, caught the disease himself. He made a partial recovery, but was never fully healthy again. He partially retired, but in 1820 wrote a scientific work on typhus based on his own experiences. • He died on May 22, 1822. Ruffini

• So, given Ruffini’s nearly correct proof of the insolvability of the quintic, why doesn’t he get the credit? – Partly timing –the mathematical world (including Lagrange) still believed it could be solved. – Partly the newness of his methods – permutation groups. – Partly the difficulty of his writing and explanation.