Unsolvable Equation

1 Évariste Galois and Niels Henrik Abel Niels Henrik Abel (1802-1829) • Born on August 5, 1802, the son of a pastor and parliament member in the small Norwegian town of Finnø. Father died an alcoholic when Abel was 18, leaving behind nine children (Abel was second oldest) and a widow who turned to alcohol • Abel was shy, melancholy, and depressed by poverty and apparent failure • Completely self-taught, Abel entered University of Christiania in 1822 • Norway separated from Denmark when Abel was 12, but remained under Swedish control. Only 11,000 inhabitants in Christiania at the time (a backwater) • 1823: Abel incorrectly believes that he has solved the general equation of the fifth degree • After finding an error, proves that such a solution is impossible • To save printing costs, paper is published in a pamphlet at his own cost, with the result in summary form • 1824: This result together with a paper on the integration of algebraic expressions (now known as Abelian integrals) led to him being awarded a stipend for study trip abroad • Abel hoped trip would allow him to marry fiancee who remained in Norway as governess (Christine Kemp) • Abel travels to Berlin, where he stays from September 1825 to February 1826 • Encouraged and mentored by the August Leopold Crelle, a promoter of science. • Crelle founds Journal für die reine und angewandte Mathematik (also known as Crelle’s Journal), the premiere German mathematical journal, and the very first volume includes several works by Abel • Impossibility of solving the quintic equation by radicals • Binomial series (contribution to the foundation of analysis) • One of the results is Abel’s Theorem on Continuity in complex analysis, which clarified and corrected some foundational results of Cauchy • In July 1826 went to Paris where remained until the end of the year • Isolated in Paris’s more elegant and traditional society; impossible to approach the great men of the Académie, e.g. Cauchy. Writes “Though I am in the most boisterous and lively place in the continent, I feel as though I am in a desert. I know almost nobody.” • Presented his Memoire sur une classe très étendue de fonctions transcendentes which contains “Abel’s Theorem” on integrals of algebraic functions. Viewed by many as Abel’s crowning achievement • Increasing financial worries toward the end of 1826 • 1827: Back in Norway, Abel cannot find work – In a letter begging friend for a loan, writes “I am as poor as a churchmouse … Yours, destroyed.” ! • Christmas 1828: holiday with his fiancee in the country, only socks to warm his hands • Violent illness (tuberculosis); died on April 6, 1829 “But I would not like to part from this ideal type of researcher, such as has seldom appeared in the history of mathematics, without evoking a figure from another sphere who, in spite of his totally different field, still seems related. <…> I compare his kind of productivity and his personality with Mozart's. Thus one might erect a monument to this divinely inspired mathematician like the one to Mozart in Vienna: simple and unassuming he stands there listening, while graceful angels float about, playfully bringing him inspiration from another world. ! Instead, I must mention the very different type of memorial that was in fact erected to Abel in Christiania and which must greatly disappoint anyone familiar with his nature. On a towering, steep block of granite a youthful athlete of the Byronic type steps over two greyish sacrificial victims, his direction toward the heavens. If needed be, one might take the hero to be a symbol of the human spirit, but one ponders the deeper significance of the two monsters in vain. Are they the conquered quintic equations or elliptic functions? Or the sorrows and cares of his everyday life? The pedestal of the monument bears, in immense letters, the inscription ABEL.” - Felix Klein Évariste Galois (1811-1832)

Born in Bourg-la-Reine, outside Paris in October 1811; father became mayor of Bourg-la-Reine Historical Background

• In 1814, Napoleon was forced to abdicate in favor of Louis XVIII • Frequent changes of power had polarized French society into those inspired by the ideals of the Revolution – the liberals and the republicans, and the “legitimists” (royalists) who wanted to revert to a church-dominated monarchy • In 1824, Louis XVIII died and was succeeded by his brother, who became King Charles X • 1823, age 11: enters the Parisian Collège de Louis-le-Grand as a boarder • Fall of 1827: Galois loses interest in all subjects except mathematics • Rhetoric teacher, who initially said “There is nothing in his work except strange fantasies and negligence,” concluded after the second term that “he is under the spell of the excitement of mathematics. I think it would be best for him if his parents would allow him to study nothing but this. Third trimester: “dominated by his passion for mathematics, he has totally neglected everything else.” • Unaware of Abel’s work, tries for two months to solve the quintic equation Disaster

• The École polytechnique, founded in 1794, was the best school for engineers and scientists in France at the time; visionaries like Lagrange and Laplace were at one time on the teaching staff • In June 1828, Galois tries to take the entrance exam a year earlier; given inadequate preparation, he fails the exam • In July 1829, a political scandal erupts in Bourg-la-Raine in which Galois’ father is framed; father commits suicide • In August 1829, Galois again takes the entrance exam – “This famous examination has become almost synonymous with Galileo’s questioning by the inquisition.” • The exam cannot be taken more than twice; instead, goes to the less prestigious École normale Revolution • In July 1830, the opposition party wins a landslide victory in the election. King Charles X, faced with abdication, attempts a coup d'état • Three Glorious Days – July 26: a series of ordinances suspending freedom of the press and annulling the results of the elections • Almost 4,000 people dead within three days • August 9: Duc d’Orleans crowned Louis-Philippe I as a compromise – Riots break out on the streets, and heavy fighting erupts – the students of the École polytechnique take charge of fighting around the Latin quarter – Galois and his classmates, locked away by the director with the help of military troops, are forced to miss the Revolution! – Galois becomes a passionate revolutionary • Galois writes anonymous letter to newspaper attacking the school’s director, which leads to expulsion • At a large banquet for the Society of the Friends of the People, Galois pulls a dagger out of his pocket and says “This is how I will be sworn in to Louis- Philippe”; perceived as making a threat against the king’s life, Galois is arrested but ultimately acquitted • He is later re-arrested on the way to another subversive gathering and sentenced to fifteen months in prison. • Spring of 1832: after a cholera outbreak in Paris, youngest prisoners are transferred to a clinic where Galois falls in love with Stéphanie Poterin- Bumotel, the daughter of one of the doctors • She eventually sought to distance herself from the affair, replying coldly to his love letters • A mysterious duel held on May 30, 1832, most likely over Stéphanie • The night before the duel, Galois went through his papers, making annotations; one of these annotations particularly apt for the event: “Je n’ai pas le temps” • Galois is fired at from twenty-five paces, dies a day later • Last known words (said to brother, Alfred): ! “Don’t cry, I need all my courage to die at twenty.” Mathematical Legacy of Galois

• Incredibly, Galois and Abel independently worked on closely related questions. The deepest work of Abel was on the integrals of algebraic functions in one variable (now known as Abelian integrals and generalizing elliptic integrals). Galois proved some of Abel’s results. • The most famous of their work concerns insolvability of quintics (equations of degree 5). Abel proved that the most general equation of degree 5 cannot be solved in radicals. Galois had a more ambitious goal: he found an algorithm that allows one to decide when any given equation can be solved in radicals, or, more generally, when this equation can be solved using other equations as intermediate steps. This machinery is now known as Galois Theory. Letter to Auguste Chevalier

• Galois’ major paper "Mémoire sur its conditions de resolubilité des équations par radicaux" was only published in 1846, thirteen years after Galois’s death, in Liouville's Journal de Mathématiques. It was too ambitious and revolutionary for its time. • The night before the duel Galois wrote a letter to his friend Chevalier in which he briefly explained his results and directions of the future research he would never be able to pursue. • The letter ends with the following words: “In my life I have often ventured to advance propositions of which I was not certain; but all that I have written here has been in my head for almost a year, and it is too much in my interest not to deceive myself that no one will suspect me of stating theorems for which I do not have complete proofs. ! Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of the theorems. Subsequently there will be, I hope, some people who will find it to their profit to decipher all this mess. ! Je t'embrasse avec effusion. ! E. Galois. May 29, 1832.”

19 The status of attempts at teaching Galois theory was well explained by Felix Klein. This was written 100 years ago but still very much true: ! “I would like to comment on the position of Galois theory as a subject in our universities. There is a contradiction here that should be deplored by both students and teachers. On the one hand, instructors are eager to teach Galois theory because of the brilliance of its discovery and the far-reaching nature of its results; on the other hand, this subject presents immense difficulties to the average beginner’s understanding. In most cases the sad result is that the instructors’ inspired and enthusiastic efforts make no impression on most of the audience, awaken no understanding.” ! At UMass, the year-long algebra sequence Math 411 - Math 412 only begins to unravel a complicated theory of groups, rings, and fields which is necessary to understand Galois’ breakthrough. Galois Theory

• What is Galois Theory? I’ll try to explain the main idea. • Imagine that we want to solve an equation of degree n: n n-1 n-2 x +a1x +a2x +…+an=0

All coefficients a1,a2,…,an are rational numbers.

• By the Main Theorem of Algebra, equation has n complex roots. • We want to find an expression for these roots using only four arithmetic operations and extracting roots (=“radicals”). • Possible if n=1, 2 (Babylon), n=3 (Ferro, Tartaglia), n=4 (Ferrari). • Also possible for some very special equations of degree 5, for example (x-1)(x-2)(x-3)(x-4)(x-5)=0 has five obvious roots. • But how about a typical, random equation?

21 Group of Permutations • The main player is the set of permutations of n objects.

• It is called the symmetric group and denoted by Sn. • There are 2 permutations of 2 objects, 6 permutations of 3 objects and, in general, n! permutations of n objects. • There is one special permutation, called identity permutation, which doesn’t permute anything. It is denoted by e. • The simplest permutation to visualize is a cycle. The cycle (12…k) takes 1 to 2, 2 to 3, …. k-1 to k and k back to 1.

• Sn is a group, which means that permutations can be composed (performed one after another).

• To practice, compose permutations (12)(34) and (13)(24).

22 Example: S3

• S3 can be visualized as the group of symmetries of an equilateral triangle • The cycle (ABC) is a rotation by 120 degrees • The cycle (AB) is a mirror reflection with respect to the vertical axis.

• S3 contains a subgroup called C3 of three elements: rotations (ABC) and (ACB) and the identity element e (rotation by 0 degrees) • The word subgroup means that composing two elements of the subgroup is again an element of the subgroup (indeed, composing two rotations is again a rotation) • Do three reflections (AB), (AC) and (BC) of S3 form a subgroup? 23 Cyclic and dihedral subgroups of Sn

24 Galois Group

n n-1 n-2 • What does it have to do with our equation x +a1x +a2x +…+an=0 ?

• The symmetric group Sn of permutations of n roots contains an amazing subgroup called the Galois group. It detects subtle properties of roots such as the possibility of writing them down using nested radicals. • The Galois group is defined as follows. Suppose we have a permutation of roots which sends every root xi to some root f(xi) of the same equation. We will try to extend this permutation to a function (called automorphism) which • Sends rational numbers to themselves: f(q)=q • Sends sums to sums f(a+b)=f(a)+f(b) • Sends products to products f(ab)=f(a)f(b) • If the extension of the permutation to an automorphism is possible then the permutation is in the Galois group. 25 Examples • What is the Galois group of (x-2)(x-5)=0? • How about x2-3=0? • This is a good illustration of an interesting theorem: The Galois group consists of just the identity if, and only if, all roots of the equation are rational.

26 Cyclic Galois groups

• What is the Galois group of x5-1=0 ? • Every equation with a cyclic Galois group can be solved in radicals (without nesting). This was essentially proved by Lagrange before Galois using a trick (nowadays called Lagrange resolvent).

27 Crucial idea • What is the Galois group G of x5-2=0 ? • Galois’ idea is to analyze this equation in two steps: 5 ★ Solve an ``auxilliary’’ equation x -1=0 (Galois group C4) ★ Solve original equation x5-2=0 but not over rationals, solve it instead over a larger “cyclotomic field’’, which also contains all 5-th roots of unity. In other words, consider automorphisms that preserve not only rational numbers but also fifth roots of unity (Galois group C5) • The Galois group G is a group of 20 elements which has a “normal” subgroup C5 and a “quotient group” C4

28 Galois Theorem • The equation is solvable in nested radicals if and only if its Galois group is, loosely speaking, “built from cyclic groups block-by-block”. Nowadays, groups like this are called solvable (by an obvious reason!)

• S2, S3 and S4 (and all their subgroups) are solvable. This explains why every equation of degree at most four is solvable in radicals.

29 The smallest non-solvable group

• The smallest non-solvable group (called A5) is the group of rotations of the icosahedron. It has 60 elements and can be realized as a subgroup of S5. • Equations of degree 5 with Galois group S5 or A5 are not solvable in radicals.

30 Further Reading

• M. Livio, The Equation that Couldn't be Solved! • F. Klein, Development of Mathematics in the 19th Century! • The MacTutor History of Mathematics, http://www- history.mcs.st-and.ac.uk/! • P. Pesic, Abel's Proof

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