MATH 412 Algebra-II

• Professor: Jenia Tevelev • Office: LGRT 1236 • Office hours Tu 3-4, Th 3-5 and by appointment • E-mail: tevelev@math.umass.edu • http://www.math.umass.edu/~tevelev/412_2015

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• Algebra is the language of mathematics and one should approach its study in the same way one learns a new language.

• Reading and listening are important but the main thing is to practice talking as often as possible. I'll try to provide many opportunities for that in the classroom and I also expect you to take an initiative and start new discussions.

• Don't be shy asking questions and I am always open to working through an extra example or clarifying a point of confusion from the previous lecture.

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Textbook

• Abstract algebra by Dan Saracino • We will cover rings, fields, and Galois Theory

(Sections 16-26) • I suggest that you read the textbook ahead of lectures. • I am a big proponent of "active reading" • When you read the proof, try to isolate its main ideas.

Every problem is unique but there aren't so many different proof techniques.

• For every definition, learn some examples that fit this definition and some that don’t.

• Take lecture notes3

Homework• Will be collected on days indicated in the course

schedule in the syllabus. • No late homework will be accepted • Use complete English sentences • Avoid “stream of consciousness” a-la James Joyce or

Virginia Woolf. Think about presenting every solution in the most clean, marketable way.

• Each time you use a special proof technique such as proof by induction or proof by contradiction, say so.

• Use quotations to theorems and factoids from the textbook or lecture notes.

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EXAMS AND QUIZZES

• Two in-class midterms and five 20-minute quizzes. The dates are in the course schedule

• Grade: 25% midterm I, 25% midterm II, 25% final exam, 15% HW, 10% quizzes.

• Two worst HW grades and one worst quiz grade will be dropped.

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So what is this course about?• For most people algebra is about solving equations. • Imagine that we want to solve an equation of degree n:

xn+a1xn-1+a2x

n-2+…+an=0, where all coefficients a1,a2,…,an are, say, rational numbers. • We want to find out if we can compute x1,x2,…,xn using only

four arithmetic operations and extracting roots (=“radicals”). • Possible when n=1 (cavemen), n=2 (Babylonians 2000 BCE),

n=3 (Ferro 1515, Tartaglia 1535), n=4 (Ferrari 1540). • The great quest to understand quintics took almost 300 years. • Almost every mathematician attempted to solve this.

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• Niels Abel and Évariste Galois proved in the 19th century that a general equation of degree 5 can not be solved in radicals.

• Their work contained an embryo of modern algebra, which was developed in the 19th and early 20th century by Leopold Kronecker, Richard Dedekind, David Hilbert, Emmy Nöther, and others. It found numerous applications to number theory, geometry, etc.

• Nowadays algebra is framed as the study of algebraic structures called groups, rings and fields. We will get to the Galois theory only at the very end of this course.

• Modern algebra is the union of three disciplines: Representation Theory, Algebraic Geometry, and Number Theory with many subfields.

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Before embarking on this long and journey, let’s put a human face on Algebra and marvel at amazing lives and works of Évariste Galois and NielsAbel - two spectacular comets of the mathematical sky.

• 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.

• Norway separated from Denmark when Abel was 12, but remained under Swedish control. Only 11,000 inhabitants in Christiania

Niels Henrik Abel (1802-1829)

• Abel was shy, melancholy, and depressed by poverty and apparent failure

• “As it happens, I am so constituted that I absolutely cannot, or at least only with the greatest difficulty, be alone. Then I become quite melancholy and not in the mood for work.” - Abel

• Completely self-taught, Abel entered University of Christiania in 1822

• 1823: Abel incorrectly believes that he has solved the general quintic; after finding his error, proves that such a solution is impossible

• To save on printing costs, paper is published in a pamphlet at his own cost, with the result in summary form

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• 1824: This result together with a paper on the integration of algebraic expressions 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 including Impossibility of solving the quintic equation by radicals

• 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

Évariste Galois (1811-1832)Born in Bourg-la-Reine, outside Paris in October 1811; father is a mayor

• 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

Historical Background

• 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

• 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

Revolution

– 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 Galois’ 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 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.”

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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.”

Group of Permutations• The main player is the group of permutations of n objects. • It is called the symmetric group or the group of permutations

and denoted 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).

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Galois Group

• Let’s get back to our equation xn+a1xn-1+a2xn-2+…+an=0 • The symmetric group Sn of permutations of n roots contains an

amazing subgroup called the Galois group of the equation. 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 can try to extend this permutation to a function (called automorphism) with a larger domain 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.

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Examples

• What is the Galois group of the quadratic equation (x-2)(x-5)=0?

• How about x2-3=0? This illustrates of an interesting • Theorem. The Galois group consists of just the

identity if, and only if, all roots of the equation are rational numbers.

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Galois Theorem• The first case to understand is when the Galois group is cyclic.

For example, A3 is a cyclic subgroup of S3 generated by (123). Galois shows that in this case the equation can be solved in radicals using a trick (Lagrange resolvent) discovered earlier by Lagrange.

• He shows that the equation is solvable in radicals if and only if its Galois group can be, loosely speaking, “built from cyclic groups block-by-block”. Nowadays, these groups are called solvable groups (by an obvious reason!)

• We can give a recursive definition: a group G is solvable if it contains a proper normal solvable subgroup H such that G/H is a cyclic group. Nesting of subgroups in the Galois group corresponds to nesting radicals in the formula for the solution.

• S2, S3 and S4 (and all their subgroups) are solvable. This is a deep reason why equations of degree 2,3 and 4 are solvable in radicals.

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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 is isomorphic to a subgroup of even permutations in S5.

• Equations of degree 5 with Galois group S5 or A5 are not solvable in radicals.

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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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