According to the most competent living mathematicians, Mrs. Noether was the most significant creative mathematical genius (female) ever born.

Albert Einstein

Amalia Emmy Noether (March 23, 1882 - April 14, 1935) was an outstanding German mathematician.

Emmy Noether was born in Erlangen, the eldest of four Jewish children. Her parents, mathematician Max Noether and Ida Amalia Kaufman, came from wealthy merchant families.

Noether initially studied languages, planning to become a teacher of English and French. To this end, she obtained permission to attend lectures at the University of Erlangen, where her father worked, at first as a volunteer (1900), and since 1904, when female education was allowed, she was officially enrolled. However, at the university, lectures in mathematics attracted Emmy more than any other. She became a student of the mathematician Paul Gordan, under whose guidance she defended her dissertation on the theory of invariants in 1907.

Already in 1915, Noether contributed to the development of the General Theory of Relativity; Einstein, in a letter to the world leader of mathematicians David Hilbert, expressed admiration for Noether's "insightful mathematical thinking".

In 1916, Noether moved to Göttingen, where the famous mathematicians David Hilbert and Felix Klein continued to work on the theory of relativity, and they needed Noether's knowledge in the field of invariant theory. Hilbert had a huge impact on Noether, making her a supporter of the axiomatic method. He tried to make Noether a Privatdozent at the University of Göttingen, but all his attempts failed because of the prejudices of the professors, mainly in the field of the humanities.

Emmy Noether's external career was paradoxical and will forever remain an example of outrageous inertia and inability to overcome prejudice on the part of the Prussian academic and bureaucratic bureaucracy. Her privatdocent title in 1919 was only due to the perseverance of Hilbert and Klein, after overcoming the extreme resistance of the reactionary university circles. The main formal challenge was the gender of the candidate: “How can a woman be allowed to become a Privatdozent: after all, having become a Privatdozent, she can become a professor and a member of the University Senate; Is it permissible for a woman to enter the Senate?" Hilbert's famous remark followed this statement: "Gentlemen, the Senate is not a bathhouse, why can't a woman enter there!"

The most fruitful period of Noether's scientific activity begins around 1920, when she creates a whole new direction in abstract algebra. Since 1922 she has been working as a professor at the University of Göttingen, heading an authoritative and rapidly growing scientific school.

If Emma Noether were a man, she would no doubt be invited to professorships by the best universities in the country. She also had to be content with the title of "extraordinary professor" of the University of Göttingen, which she received on April 6, 1922, when she was already forty years old. By this time, she was already rightfully considered among specialists as the founder of modern algebra, she managed to lay the cornerstones in the foundations of several important scientific areas. The decree appointing Emma Noether to the post of extraordinary professor specifically stipulated that she was not entitled to any privileges provided for by a civil servant.

Contemporaries describe Noether as an extremely intelligent, charming and affable woman. Her femininity was manifested not outwardly, but in a touching concern for her students, her constant readiness to help them and her colleagues. Among her devoted friends were world-famous scientists: Hilbert, Hermann Weyl, Edmund Landau, the Dutch mathematician L. Brouwer, Soviet mathematicians P.S. Aleksandrov, P.S. Uryson and many others.

In 1924-1925, Emmy Noether's school made one of its most brilliant acquisitions: Barthel Leendert van der Waerden, a graduate from Amsterdam, became her pupil. He was then in his 22nd year, and this was one of the brightest young mathematical talents in Europe. Van der Waerden quickly mastered the theories of Emmy Noether, supplemented them with significant new results, and, like no one else, contributed to the dissemination of her ideas. The course on the general theory of ideals given by van der Waerden in 1927 in Göttingen was an enormous success. The ideas of Emmy Noether, brilliantly expounded by van der Waerden, conquered mathematical public opinion, first in Göttingen and then in other leading mathematical centers in Europe.

Basically, Noether's works relate to algebra, where they contributed to the creation of a new direction, known as abstract algebra. Noether made a decisive contribution to this field (along with Emil Artin and her student van der Waerden).

The terms "Noetherian ring", "Noetherian module", normalization theorems and the Lasker-Noether ideal decomposition theorem are now fundamental.

Noether made a great contribution to mathematical physics, where the fundamental theorem of theoretical physics (published in 1918) is named after her, linking conservation laws with system symmetries (for example, the homogeneity of time entails the law of conservation of energy). This fruitful approach is the basis of the famous series of books "Theoretical Physics" by Landau-Lifshitz. Noether's theorem is of great importance in quantum field theory, where the conservation laws arising from the existence of a certain symmetry group are usually the main source of information about the properties of the objects under study.

Noether's ideas and scientific views had a huge impact on many scientists, mathematicians and physicists. She raised a number of students who became world-class scientists and continued the new directions discovered by Noether.

Noether adhered to social democratic views. For 10 years of her life she collaborated with mathematicians of the USSR; in the 1928-1929 academic year, she came to the USSR and lectured at Moscow University, where she influenced L.S. Pontryagin and especially on P.S. Alexandrov, who had often visited Göttingen before.

Since 1927, the influence of Emmy Noether's ideas on modern mathematics has been growing all the time, and in parallel, the scientific fame of the author of these ideas has also increased. If in 1923-1925 she had to prove the importance of the theories she developed, then in 1932, at the international mathematical congress in Zurich, she was crowned with the laurels of the most brilliant success. Noether, together with his student Emil Artin, receives the Ackermann-Thöbner Prize for achievements in mathematics. The large review report she read at this congress was a real triumph of the direction she represented, and she could not only with inner satisfaction, but also with the consciousness of unconditional and complete recognition, look back at the mathematical path she had traveled. The Zurich Congress was the high point of her international scientific position. A few months later, a catastrophe broke out over German culture, and in particular over that center of culture that had been the University of Göttingen for centuries.

In 1933, Hitler came to power in Germany and the German government passed the Civil Service Law. The idea behind this law was simple: "Get out of the non-Aryans!" The teachers in Germany were civil servants, and the idea about them was simply: "Aryan students should be taught by Aryan professors."

Emmy Noether was among the first six professors to be banned from lectures by the Prussian Ministry and sent on indefinite leave under the infamous law that initiated the massive purge of the teaching staff.

Personally, Noether received an official paper signed by the head of the Prussian Ministry of Science, Art and Public Education in April 1933. It was written in plain text: "In accordance with paragraph 3 of the Civil Service Code of April 7, 1933, I deprive you of the right to teach at the University of Göttingen."

One of the greatest tragedies of all experienced by human culture since the time of the renaissance occurred, a tragedy that a few years ago seemed incredible and impossible in Europe of the 20th century. One of its many victims was the Göttingen algebraic school founded by Emmy Noether: its leader was expelled from the university walls; having lost the right to teach, Emmy Noether had to emigrate from Germany.

Emmy's younger brother, the gifted mathematician Fritz Noether, left for the USSR, where he was shot in September 1941 for "anti-Soviet sentiments."

Even after leaving Germany, Emma Noether showed no trace of bitterness or enmity towards those who ruined her life. She turned out to be one of the few emigrants who, the very next year after her departure, dared to return: in the summer of 1934, she decided to spend some time in the familiar surroundings of the green Göttingen, where she had worked so well all the last years.

In exile, Emma faced the same difficulties as most other scientists who came overseas already in adulthood. But she managed to find a job relatively quickly. She received a teaching position at the small American college Bryn Mawr in Pennsylvania and did research work at the Institute for Advanced Study in Princeton.

Having settled down on her own, she immediately began to take care of colleagues who were less fortunate in exile. Together with Hermann Weyl, she organized a special "Foundation for helping German mathematicians", to which those scientists who had already found work were to deduct a small part of their salary. Scholarships were paid out of the collected funds to those who were in particular need of support.

And in America, not everyone understood the scale of her personality as a scientist and a person. The records of the Daggen Emergency Committee preserved an entry made on March 21, 1935, three weeks before the unexpected death of the brilliant scientist: “Yesterday there was a discussion with the president of Bryn Mawr College about the fate of Emmy Noether. She said that Emma Noether was too eccentric and difficult to adapt to American conditions to get a permanent contract with her, but she would keep her in college for another two years.

Unfortunately, Emma was not allowed to work in college for these two years: on April 14, 1935, after an unsuccessful medical operation to remove a cancerous tumor, she died.

In his speech, the President of the Moscow Mathematical Society P.S. Alexandrov, at a meeting of the society on September 5, 1935, began with the following words:

On April 14 this year in the small town of Bryn Mawr (USA, Pennsylvania), after a surgical operation, Emmy Noether, one of the greatest mathematicians of our time, a former professor at the University of Göttingen, died at the age of 53. The death of Emmy Noether is not only a great loss for mathematics,It is a tragic loss in the full sense of the word. In the peak of her creative powers, the greatest female mathematician who ever existed died, she died, expelled from her homeland, cut off from her school, which she had been creating for years and was one of the most brilliant mathematical schools in Europe, she died cut off from her relatives, who turned out to be scattered over different countries by virtue of the same political barbarity by virtue of which she herself had to emigrate from Germany. The Moscow Mathematical Society today mournfully bows before the memory of one of its most outstanding members, who continuously for over ten years maintained close ties of constant scientific interaction, sincere sympathy and cordial friendship with the society, with mathematical Moscow and with the mathematicians of the Soviet Union ...

Emmy Noether is named after:

• crater on the moon
• asteroid
• street in Noether's hometown, Erlangen
• the school where she studied in Erlangen.
• German program to support outstanding young scientists: Emmy Noether Program.

The following mathematical objects bear the Noether name:

• Noetherian ring
• noether module
• Noether's theorem
• Lasker-Noether theorem
• Skolem-Noether theorem
• noetherian spaces
• Noetherian scheme
• noether problems
• Noether's lemma.

Based on materials from Wikipedia and sites: berkovich-zametki.com and turtle-t.livejournal.com, as well as articles by P.S. Aleksandrov, “In memory of Emmy Noether” (Usp.

Amalia (Emmy) Noether, queen without a crown

According to the most eminent living mathematicians, Emmy Noether was the greatest creative mathematical genius to appear in the world since higher education was opened to women.

Albert Einstein

Einstein was right and Emmy Noether (1882–1935) , with whom he never got to work together at the Institute for Advanced Study at Princeton (although she deserved it more than anyone), was an amazing mathematician - perhaps the greatest woman mathematician of all time. And Einstein was not alone in this view: Norbert Wiener placed Noether on a par with two-time Nobel Prize winner Marie Curie, who was also an excellent mathematician.

Also, Emmy Noether became the object of a number of bad jokes - let us recall at least the immortal phrase of the intemperate language of Edmund Landau: "I can believe in her mathematical genius, but I cannot swear that this is a woman." Emmy was indeed masculine in appearance, and besides that, she did not think at all about how she looked, especially during classes or scientific debates.

According to eyewitnesses, she forgot to style her hair, clean her dress, chew her food thoroughly, and was distinguished by many other features that made her not too feminine in the eyes of decent German compatriots. Emmy also suffered from severe myopia, which is why she wore ugly glasses with thick glasses and looked like an owl. To this should be added the habit of wearing (for reasons of convenience) a man's hat and a leather suitcase stuffed with papers, like an insurance agent. Hermann Weyl himself, a student of Emmy and an admirer of her mathematical talent, quite balancedly expressed the general opinion about the mentor with the words: "The graces did not stand at her cradle."

Portrait Emmy Noether in youth.

Transformation into a beautiful swan

Emmy Noether was born into a society where women were, one might say, shackled hand and foot. At that time, the all-powerful Kaiser Wilhelm II, a lover of solemn receptions and ceremonies, ruled in Germany. He came to the city, decorously descended from the train, and then the local mayor made a speech. All the dirty work was done by Iron Chancellor Bismarck. He was the true head of state and society, the inspirer of his conservative structure, which prevented the education of women (universal education was considered a sign of hated socialism). The model for a woman was the Kaiser's wife, Empress Augusta Victoria. Her life credo was four K: Kaiser, kinder(children), Kirche(church), K?che(kitchen) - an augmented version of the three K from the folk trilogy " Kinder, Kirche, K?che". In such an environment, women were assigned a clearly defined role: on the social ladder they were below men and one step above domestic animals. So, women could not get an education. Actually, the education of women was not completely prohibited - for the homeland of Goethe and Beethoven, this would be too much. Overcoming many obstacles, women could study, but were not eligible to hold positions. The result was the same, but the game was more subtle. Some teachers, demonstrating a special ideological zeal, refused to start classes if at least one woman was present in the audience. The situation was quite different, for example, in France, where freedom and liberalism dominated.

Emmy was born in the small town of Erlangen to an upper-middle-class teaching family. Erlangen occupied an unusual place in the history of mathematics - it was the small birthplace of the creator of the so-called synthetic geometry Christian von Staudt (1798–1867) moreover, it was in Erlangen that the young genius Felix Klein (1849–1925) published his famous Erlangen program, in which he classified geometries from the point of view of group theory.

Emmy's father, Max Noether, taught mathematics at the University of Erlangen. His intelligence was inherited by his son Fritz, who devoted his life to applied mathematics, and his daughter Emmy, who resembled the ugly duckling from Andersen's fairy tale - no one could even imagine what scientific heights she would reach. In childhood and adolescence, Emmy was no different from her peers: she really liked to dance, so she willingly attended all the celebrations. At the same time, the girl did not show much interest in music, which distinguishes her from other mathematicians, who often love music and even play different instruments. Emmy professed Judaism - at that time this circumstance was unimportant, but it affected her future fate. With the exception of occasional flashes of genius, Emmy's education was no different from that of her peers: she knew how to cook and manage a household, showed success in learning French and English, and she was prophesied a career as a language teacher. To everyone's surprise, Emmy chose math.

Facade of the Kollegienhaus - one of the oldest buildings of the University of Erlangen.

Endless Race

Emmy had everything she needed to devote herself to her chosen occupation: she knew mathematics, her family could allocate funds for her life (albeit very meager ones), and personal acquaintance with her father's colleagues allowed her to rely on the fact that studying at the university would not become unbearable . To continue her studies, Emmy had to become a student - she was forbidden to attend classes as a full student. She successfully completed her studies and passed the exam that gave her the right to receive a doctoral degree. Emmy chose algebraic invariants of ternary quadratic forms as her dissertation topic. The teacher of this discipline was Paul Gordan (1837–1912) , whom contemporaries called the king of invariant theory; he was a longtime friend of Noether's father and a supporter of constructive mathematics. In his search for algebraic invariants, Gordan turned into a real bulldog: he clung to an invariant and did not open his jaws until he singled it out among the intricacies of calculations that sometimes seemed endless. It is not too difficult to explain what an algebraic invariant and form are, but these concepts are not of interest to modern algebra, so we will not dwell on them in more detail.

In his doctoral dissertation entitled "On the definition of formal systems of ternary biquadratic forms", 331 invariants of ternary biquadratic forms found by Emmy are given. The work earned her a doctorate and gave her plenty of practice in mathematical gymnastics. Emmy herself later, in a fit of self-criticism, called this hard work nonsense. She became the second female doctor of science in Germany after Sofia Kovalevskaya.

Emmy got a teaching position in Erlangen, where she worked for eight long years without any salary. Sometimes she had the honor to replace her own father - his health had weakened by that time. Paul Gordan retired and was replaced by Ernst Fischer, who was more modern and got on well with Emmy. It was Fischer who introduced her to the works of Hilbert.

Fortunately, Noether's insight, her mind and knowledge were noticed by two luminaries of the University of Göttingen, "the most mathematical university in the world." These luminaries were Felix Klein and David Gilbert (1862–1943) . It was 1915, the First World War was in full swing. Both Klein and Hilbert were extremely liberal in the education of women (and their participation in research work) and were specialists of the highest level. They persuaded Emmy to leave Erlangen and move in with them in Göttingen to work together. At the time, Albert Einstein's revolutionary physics ideas were booming, and Emmy was an expert on algebraic and other invariants, which constituted an extremely useful mathematical apparatus of Einstein's theory (we will return to the discussion of invariants a little later).

All this would be funny if it were not so sad - even the support of such authorities did not help Emmy overcome the resistance of the Academic Council of the University of Göttingen, from whose members one could hear statements in the spirit: “What will our heroic soldiers say when they return to their homeland, and in auditoriums, will they have to sit in front of a woman who will address them from the pulpit?” Hilbert, who was present at such a conversation, objected indignantly: “I don’t understand how the gender of the candidate prevents her from being elected Privatdozent. After all, this is a university, not a men's bath!

But Emmy was never elected Privatdozent. The Academic Council declared a real war on her. The conflict soon ended, the Weimar Republic was proclaimed, and the situation of women improved: they got the right to vote, Emmy was able to take the position of professor (but without salary), but only in 1922, with great efforts, she finally began to receive money for her work. Emmy was annoyed that her time-consuming work as editor of the Annals of Mathematics was not appreciated.

In 1918 Noether's sensational theorem was published. Many called it that, although Emmy proved many other theorems, including very important ones. Noether would have deserved immortality even if she died the day after the theorem was published in 1918, even though she had actually found the proof three years earlier. This theorem does not belong to abstract algebra and is located at the interface between physics and mathematics, more precisely, belongs to mechanics. Unfortunately, in order to explain it in a language understandable to the reader, even if in a simplified form, we cannot do without higher mathematics and physics.

Speaking simply, without symbols and equations, Noether's theorem in the most general formulation says: "If a physical system has continuous symmetry, then there are corresponding quantities in it that retain their values ​​over time."

The concept of continuous symmetry in higher physics is explained with the help of Lie groups. We will not go into details and say that in physics, symmetry is understood as any change in a physical system with respect to which the physical quantities in the system are invariant. This change, by means of a mathematically continuous transformation, must affect the coordinates of the system, and the quantity under consideration must remain unchanged before and after the transformation.

Where did the term "symmetry" come from? It belongs to a purely physical language and is used because it is similar in meaning to the term "symmetry" in mathematics. Imagine rotations of space forming a symmetry group. If we apply one of these rotations to a coordinate system, we get a different coordinate system. Change of coordinates will be described by continuous equations. According to Noether's theorem, if a system is invariant with respect to such a continuous symmetry (in this case, rotation), then it automatically has a conservation law for one or another physical quantity. In our case, after carrying out the necessary calculations, we can make sure that this value will be the angular momentum.

We will not dwell on this topic and give some varieties of symmetry, symmetry groups and the corresponding physical quantities that will be preserved.

This theorem has received many accolades, including from Einstein, who wrote to Hilbert:

« Yesterday I received a very interesting article by Mrs. Noether on the construction of invariants. I am impressed that such things can be considered from such a general point of view. It would do no harm to the old guard in Göttingen if they were sent to be trained by Madame Noether. Looks like she knows her trade well».

The praise was well-deserved: Noether's theorem played a non-trivial role in solving problems in the general theory of relativity. This theorem, according to many experts, is fundamental, and some even put it on a par with the well-known Pythagorean theorem.

Fast forward to a simple and understandable world of experiments, described Karl Popper (1902–1994) , and suppose that we have created a new theory describing some physical phenomenon. According to Noether's theorem, if there is some kind of symmetry in our theory (it is quite reasonable to assume such a thing), then some quantity that can be measured will remain in the system. In this way, we can determine whether our theory is correct or not.

THEOREM NOETHER

A physical system in mechanics is defined using fairly complex terms, including such a concept as an action, which can be considered as the product of the released energy and the time spent on its absorption. The behavior of a physical system in the language of mathematics is described by its Lagrangian L, which is a functional (function of functions) of the form

where q- position, q?- speed (the dot at the top in Newton's notation denotes the derivative of q), t- time. note that q- position in a general coordinate system, which is not necessarily Cartesian.

Action BUT in the language of mathematics is expressed by an integral along the path chosen by the system:

The principle of least action, which played such an important role in 19th century physics, states that a physical system moves according to the law of least effort, therefore, if we use the language of mathematical analysis, the action A must be an extreme value, that is, a minimum or maximum, so its first derivative must equal zero.

A good illustration is worth a thousand words, so here is an example that is perfectly explained in many books and on the Internet. Noether's theorem in this example is expressed as follows: "Let's assume that the system of particles has some symmetry, that is, its Lagrangian L invariant under changes in some variable s so that dL/ds= 0. Then there is a property of the system With, which will be saved: DC/dt = 0

Consider a physical system consisting of two springs with elasticity coefficients to 12 and to 23 Let us introduce the notation:

Now consider the symmetry (in the formulation of the theorem, it is denoted by s). Since the law of elasticity is always satisfied, we may well assume that s = t, that is, time, and the symmetry of the Lagrangian, which is mentioned in the original formulation, manifests itself as follows:

Let's carry out some algebraic transformations:

Let's change the order of the members:

We have obtained the conserved quantity With- it is given in parentheses. As q? = X?, we have

The sum (with a minus sign) of kinetic and potential energy, that is, the total energy of the system, is constant. We have received the law of conservation of energy.

Algebra and more algebra. And what algebra!

We interrupted our story about Emmy with the fact that she settled in Göttingen, next to Klein and Hilbert, two world-famous mathematicians. The witty Gilbert found a way to overcome obstacles from the most inert and conservative teachers: he organized courses under his own name, but Emmy replaced him in the classroom every time, and ill-wishers could only gnash their teeth.

Emmy was distinguished by her incredible performance - she could be compared with a car whose brakes failed. In 1920, she decided to follow a new path. Gradually, but steadily, Emmy began to pay more and more attention to questions of pure algebra: first, rings and ideals on rings, then more complex structures, in particular, various algebras. She mastered the topic so much that she fully deserved the title of "Lord of the Rings." Such important results for the development of algebra as the Lasker-Noether theorem (1921) and the normalization lemma (1926) belong to this era. By 1927, her isomorphism theorems date back.

Then, almost immediately, Emmy moved on to more complex topics, in particular algebra. In 1931, the Albert-Brauer-Hasse-Noether theorem on algebras of finite dimension was formulated. In 1933, Emmy Noether again obtained an important result related to algebras, the so-called Skolem-Noether theorem. We do not give detailed formulations of these theorems, since they mention very abstract mathematical terms and objects that are available only to specialists.

Emmy was followed everywhere by a real crowd of students - noisy, unruly, but very smart. These were the "children of Noether" who listened to her words. They accompanied her on long walks and frequent swims in the municipal swimming pool, where Emmy swam and dived like a dolphin. Many "Noether children" later became great mathematicians thanks to the ideas they learned from their mentor, although her pedagogical gift was, so to speak, non-standard: she treated her students like a mother hen to chickens - she was invariably strict and demanding and not she didn't step aside from them. To many, she looked more like a rooster than a chicken, and they called her, showing respect for her mind and some timidity, in the masculine gender - Der Noether.

"Children Noether».

To understand how curious the retinue of the “children of Noether” was, an anecdotal case from the time of Nazi Germany will help. Natasha Artin-Braunschweig, wife Emil Artina (1898–1962) , told how they once went down to the Hamburg metro: the students did not lag behind Noether and followed her like children behind the Pied Piper of Hamelin. As soon as they got on the train, Emmy began to discuss mathematical topics with Emil Artin, raising her voice more and more and not paying attention to the other passengers. In Noether's speech, the words "fuhrer" and "ideal" were constantly heard - to the great horror of Natasha, who was afraid that they were about to be detained by the Gestapo.

However, any of the "children" could easily explain to the terrifying Gestapo that these words were just innocent algebraic terms from the theory of rings. At that time, the Nazis installed rampant surveillance, they interfered in the private lives of people and literally besieged universities. One of Emmy's students, who was Jewish and therefore unable to attend university, came to study at her home in the form of a member of the assault squad to avoid suspicion. The pacifist Emmy perceived what was happening with humility.

She was engaged in the most modern sections of algebra. From time to time, Emmy turned to topology, in particular in collaboration with Pavel Sergeevich Alexandrov (1896–1982) . Noether's specialty was the detailed study of algebraic structures, the purpose of which was to discard their particular properties and consider them in the most general way possible. Emmy enjoyed unlimited authority, and students came to her from all over Europe. One of them, Barthel van der Waerden (1903–1996) , who later became famous as the author of "Modern Algebra", a book that became the canon for several generations (from this very book, the pages of which were dotted with incomprehensible symbols of the Gothic type, I also studied), wrote in Emmy Noether's obituary:

« For Emmy Noether, the connections between numbers, functions and operations became clear, generalizable and useful only after they were separated from specific objects and reduced to general conceptual connections.».

Here is what Einstein wrote:

« Theoretical mathematics is a kind of poetry of logical ideas. Its goal is to search for the most general ideas that describe the maximum possible range of formal relationships in a simple, logical and general way. On this path to logical beauty, we discover formulas that allow us to better understand the laws of nature.».

Basic algebraic structures

Read this section on the basics of abstract algebra carefully, otherwise you will not understand anything of what is said in the following sections. This section is extensive but simple as it contains only definitions.

There are many basic algebraic structures that are considered as sets with one or more operations. We confine ourselves to considering structures on which two operations are defined, o and . These operations are often + and . Sometimes the so-called third law of external composition is required ( a sometimes more), but we will consider only the simplest cases. Instead of constantly using the words "is an element", we will replace them with the symbol

.

A group is a set of elements BUT with an operation o defined on it that satisfies the following three conditions:

1) there is a neutral element n such that n about a = a about n = a for anyone a

2) for each a

BUT there is an inverse element a-1 such that a about a -1 = a-1 about a = n;

3) for any a, b, c

BUT the associativity property holds, according to which ( a about b) about with= a about ( b about with).

A group is called commutative, or Abelian (in honor of the Norwegian mathematician Niels Henrik Abel), if for any a, b

BUT the operation we have defined is commutative, that is, the relation a about b = b about a.

If the operation of addition (+) is defined on the group, then the element inverse a, denoted - a and is called opposite. The neutral element in this case is denoted 0.

If the operation of multiplication () is defined on the group, then the element inverse a, denoted by 1/ a. The neutral element in this case is denoted 1.

4) for any a, b, c

And it's fair ( a b) with = a (b c).

The operations o and are related to each other by the property of distributivity with respect to:

5) a (b about with) = (a b) about ( a c).

A ring is a commutative group on which one more operation is defined that has the associativity property:

Examples of rings are natural numbers

Whole numbers

Rational numbers

Real numbers

And complex numbers

(regardless of the modal arithmetic defined for them). Polynomials also form rings.

Operation in the world of rings about has commutativity similar to the operation of addition, so it is denoted by the sign +. The operation (for simplicity, we will assume that it also has commutativity) is denoted by the symbol · , like multiplication.

Subgroup or subring BUT will be any subset that will remain a group or ring if the operations are restricted about or this subset. The ideal is a special subring: this subring AT

BUT such that any work b AT and any other element that belongs to AT or not, will belong AT. Ideals can be added and multiplied. The results of addition and multiplication of ideals will also be ideals. The concept of ideal arose as a generalization of the concept of number. For two given ideals I and J we have:

Define the ideal IJ somewhat more difficult. This is the ideal generated by all works hu, where X

I, y J. The intersection of all ideals containing similar products is called a generated ideal.

The area of ​​integrity is called the ring BUT, on which for the operation · there are no so-called zero divisors. In other words, there are no elements on this ring a and b such that ab = ba= 0.

In this case, the ring BUT is commutative and contains an identity element, that is, a neutral element is defined for the operation, which plays the role of a unit:

a 1 = a.

Now consider the area of ​​integrity BUT without 0. Denote it by BUT* = BUT|(0). If the operation · determines on BUT* commutative group, then BUT called a field. If a BUT* is not commutative, then BUT called the body. Do not be afraid of such difficulties: if the ring BUT of course, then it is commutative by the famous Wedderburn theorem. If the ring BUT infinitely, then there is freedom for algebraists.

Let's consider A-modules - the rarest type of the modern algebraic world. To define a left A-module, we need a ring with identity BUT and the commutative group M. Actions with elements a, b

BUT and elements M (m,n M) are defined in the following way:

1. (ab)m= a(bm)

2. (a + b) n = am + bm

3. a(m + n) = am + an

4. 1m = m.

The right A-module is defined similarly; a commutative module (or simply an A-module) is a module that is right and left at the same time. If A is a field, then the A-module is called a vector space. If the operation of multiplication is defined for the vectors of a vector space, we have an "algebra". This is where we will stop. Although the definitions we have given are elementary, it is quite possible that the reader will not call this section elementary.

A few words about algebra, ideals and Noetherian rings

Most of Emmy Noether's scientific work was devoted to rings and ideals - algebraic structures, on which she worked for many years. Why did Noether pay them such attention?

Many objects that mathematicians work with are rings: for example, rings are the set of integers

And its successive extensions are ,

Rings are also polynomials of one variable with coefficients from the above rings

[X]. Similarly, rings are polynomials of several variables

As well as convergent series - in short, a lot more.

But what are ideals and why did they get such a romantic name? Let's make a small digression into the history of mathematics. Consider as an example the quadratic integer

[?-5] or

Which is similar. This is a set of numbers like a + b?-5, where a and b- whole numbers. In other words,

[?-5] is a ring (check it out), but here, mathematically speaking, we are entering a forbidden zone. We are accustomed to the standard properties of divisibility and to the fact that the factorization of a number into prime factors is always unique. For example, consider the number 21. We have 21 = 3 7 and this is where the factorization ends: 21 can be factored into prime factors in a unique way, and these factors will be 3 and 7. This statement follows from the main theorem of arithmetic: on the set

The decomposition of any number into prime factors is unique. On the set

[?-5] this statement will no longer hold: here we can factorize 21 into prime factors in two ways:

3 7 \u003d (4 + ?-5) (4 - ?-5) \u003d 21.

On this set, the decomposition into prime factors will no longer be the only one, which, to his greatest displeasure, was noticed by Ernst Kummer (1810–1893) . This statement, which seems not very important and is written in just one line, prevented the algebraists of the XIX century from proving Fermat's theorem and gave them a lot of trouble.

In order to somehow correct the situation and get around the problem, Kummer himself introduced ideal numbers. They were not very useful, since they no longer belonged to

[?-5], but to another, larger ring. These were not even numbers - today we would call them sets of numbers that are equivalent to each other. Mathematicians of that time were unaware of the currently generally accepted concepts of set factor and homomorphism, and it was only Richard Dedekind (1831–1916) . He was followed by other algebraists who cleared the area and began excavations. Emmy Noether occupied an important place among them.

Ideals have another remarkable feature - we are talking about a chain of ideals. We will not follow Noether and try to explain an abstract concept, but we will limit ourselves to giving one very simple example - the ideals of the ring of integers

.

In this world (it is an area of ​​integrity, that is, a “good” ring), the main theorem of arithmetic rules the show: for all numbers, decomposition into prime factors is unique, and nothing disturbs harmony. The ideals in this world will be multitudes n

Consisting of integer multiples n. The number of such ideals, as well as the numbers themselves, will be infinitely large. The sum and product of ideals are defined very simply:

Ideals, which are sets of numbers, and ordinary numbers behave in the same way, are factored in the same way, and are equivalent from the point of view of arithmetic. They are equivalent even in such a difficult aspect as divisibility. Indeed, " b divided by a» for ideals can be expressed as b

Noether's genius lies in the fact that she built a chain of ideals, united by a membership function

Which reflects their divisibility to each other.

Since any divisibility relation sooner or later ends with a certain number, sooner or later any chain of ideals will also end. "Good" chains of ideals necessarily end, that is, they are finite. Rings on which there are no infinite chains of ideals are called Noetherian rings. It was these rings that Emmy paid special attention to in her research.

Later, algebraists proved the equivalence of the following statements.

1. Ring BUT is Noetherian (in other words, increasing chains of ideals on it are finite).

2. Any ideal on BUT is finitely generated.

3. Any set of ideals on BUT contains the greatest ideal.

In 1999, the Australian Mathematical Foundation produced T-shirts featuring ever-increasing chains for the ideal 18

On the set

The limited size of T-shirts prevented us from using another example. The following chains of ideals were depicted on the T-shirts:

As expected, these chains are finite, and the ring

Is Noetherian. By the way, Hilbert proved that if the ring A is Noetherian, then the polynomial ring will also be Noetherian. BUT[X].

THEOREM EMMI AND CHESS PLAYER

Algebraist Emanuel Lasker (1868–1941) was an outstanding mathematician and world chess champion. He considered in detail the ordinary, simple and primary ideals. We will not delve too deeply into abstract algebra and consider the rings BUT, which are also integrity regions. An approximate ideal on these rings is called the ideal I, different from the original ring BUT, on which ab

I and a I exist n such that b n I. (At n= 1 this ideal is called simple.) Lasker described a very wide class of rings (today they are called Lasker rings) based on one interesting property of their ideals. Any ideal can be represented as the intersection of a finite number of primary ideals.

Emmy Noether proved a theorem known today as the Noether-Lasker theorem, which reads as follows:

"Any Noetherian integrity domain is a Lasker ring."

This theorem, related to abstract algebra, connects two seemingly very distant concepts - finite chains of ideals and intersections of primary ideals. You may not have noticed (and, in fact, you should not apologize for this at all) that if we apply the Lasker-Noether theorem to the ring

Then we get the basic theorem of arithmetic: any integer can be represented as a product of prime factors in a unique way. The term "Noetherian ring", which is used everywhere today, was introduced by the great French mathematician Claude Chevalley (1909–1984) , one of the founders of the Bourbaki group.

End of story

Needless to say, already in the 1930s, Emmy Noether enjoyed incredible respect among mathematicians. An example of this is her participation in the International Congress of 1932. The following year, the Nazis came to power in Germany, and with great determination, which could only be compared with their own stupidity, they began to expel all Jewish teachers from universities. Emmy also suffered from anti-Semitism. Her friends and acquaintances protested in vain - she and many of her colleagues (Thomas Mann, Albert Einstein, Stefan Zweig, Sigmund Freud, Max Born and others) were forced to stop teaching in Germany and leave the country (as it became clear later, not everyone had such an opportunity ) to spread their evil ideas among members of other, non-Aryan races. What exactly the Nazis saw as harmful in modern algebra, we will never know. Most likely, the Nazis themselves did not know the answer to this question.

Emmy's brother, Fritz, moved to Tomsk, and Emmy herself, who for some time leaned either to Oxford or Moscow (she had a certain sympathy for the socialist revolution in the USSR), ended up in the United States through the efforts of the Rockefeller Foundation.

Many books have been written about anti-Semitism and its spread. It would be useful to say that prior to the entry of the United States into World War II, anti-Semitism was gaining momentum in some universities that were considered temples of knowledge and strongholds of liberalism, in particular at Princeton University in New Jersey. It is for this reason that the Jewish family of millionaires and philanthropists, the Bambergers, donated several million dollars to the Institute for Advanced Study in Princeton, an absolutely neutral institution free from such prejudices. This donation ultimately helped the institute become a model research institution. At Princeton, scientists hatched ideas, were paid solely for scientific work, and were exempted from teaching. The institute became a haven for many European emigrants who were wholly or half Jewish. Among them were Einstein, Weyl, von Neumann and Gödel. Although Emmy Noether lectured at the institute and conducted seminars, and her achievements in mathematics were more than enough, she never became a full-fledged employee of Princeton - only because she was a woman. Noether's main place of work was Bryn Mawr College located near New Jersey in Pennsylvania - the best women's college in the world. Emmy sometimes forgot that she was in America, and in the midst of an argument about mathematics, she burst into rants in German.

Just two years after arriving in America, doctors discovered Emmy had uterine cancer. She had an excellent operation, but died of an embolism. Interestingly, among the avalanche of obituaries, one, signed by van der Waerden, was published in Germany without much problem - the Nazi censors must not have been very good at algebra.

A crater on the far side of the Moon and asteroid 7001 are also named after Emmy Noether.

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Prominent German mathematician, "the largest female mathematician that ever existed."

Born in the family of mathematician Max Noether in Erlangen. She studied at the University of Erlangen, where her father worked, at first as a volunteer, since 1904, when female education was allowed, she was officially enrolled. She was a student of the mathematician Paul Gordan, under whose guidance she defended her dissertation on the theory of invariants in 1907.

Already in 1915, Noether contributed to the development of the General Theory of Relativity; Einstein, in a letter to the world leader of mathematicians David Hilbert, expressed admiration for Noether's "insightful mathematical thinking".

In 1916, Noether moved to Göttingen, where the famous mathematicians David Hilbert and Felix Klein continued to work on the theory of relativity, and they needed Noether's knowledge in the field of invariant theory. Hilbert had a huge impact on Noether, making her a supporter of the axiomatic method. He tried to make Noether a Privatdozent at the University of Göttingen, but all his attempts failed because of the prejudices of the professors, mostly humanists. Hilbert's phrase became known:

I don't understand why the candidate's gender serves as an argument against electing her as Privatdozent. After all, this is a university, not a men's bath!

Noether, however, without holding any office, often lectured for Hilbert. Only after the end of World War I was she able to become a Privatdozent in 1919, then (1922) a supernumerary professor.

The most fruitful period of Noether's scientific activity begins around 1920, when she creates a whole new direction in abstract algebra. Since 1922 she has been working as a professor at the University of Göttingen, heading an authoritative and rapidly growing scientific school.

Contemporaries describe Noether as not very beautiful, but extremely intelligent, charming and friendly woman. Her femininity was manifested not outwardly, but in a touching concern for her students, her constant readiness to help them and her colleagues. Among her devoted friends were world-famous scientists: Hilbert, Hermann Weyl, Edmund Landau, the Dutch mathematician L. Brouwer, Soviet mathematicians P. S. Aleksandrov, P. S. Uryson and many others.

Noether adhered to social democratic views. For 10 years of her life she collaborated with mathematicians of the USSR; in the 1928/29 academic year, she lectured at Moscow University, where she influenced L. S. Pontryagin and especially P. S. Aleksandrov, who had often visited Göttingen before. P. S. Alexandrov recalled:

Emmy Noether's lectures on the general theory of ideals were the pinnacle of everything I heard that summer in Göttingen... Of course, Dedekind laid the very beginning of the theory, but only the very beginning: the theory of ideals in all the richness of its ideas and facts, a theory that had such a huge impact on modern mathematics , is the creation of Emmy Noether. I can judge this because I know both Dedekind's work and Noether's main works on ideal theory.

Noether's lectures captivated both me and Urysohn. They were not brilliant in form, but they conquered us with the richness of their content. We constantly saw Emmy Noether in a relaxed atmosphere and talked a lot with her, both on the topics of the theory of ideals and on the topics of our work, which immediately interested her.

Our acquaintance, which began vividly this summer, deepened very much the following summer, and then, after Urysohn's death, turned into that deep mathematical and personal friendship that existed between Emmy Noether and me until the end of her life. The last manifestation of this friendship on my part was a speech in memory of Emmy Noether at a meeting of the Moscow International Topological Conference in August 1935.

1932: Noether, together with Emil Artin, receives the Ackermann-Töbner Prize for achievements in mathematics.

After the Nazis came to power in 1933, Noether, as a Jew, had to emigrate to the United States, where she became a teacher at the women's college in Bryn Mawr (Pennsylvania) and a visiting professor at the Institute for Advanced Studies in Princeton. Emmy's younger brother, the gifted mathematician Fritz Noether, left for the USSR, where he was shot in September 1941 for "anti-Soviet sentiments."

Despite brilliant mathematical achievements, Noether's personal life did not work out. Being an ugly woman, she never married. Non-recognition, exile, loneliness in a foreign land, it would seem, should have spoiled her character. However, she almost always appeared calm and benevolent. Hermann Weil wrote that even happy.

Emmy Noether died in 1935 after an unsuccessful operation to remove a cancerous tumor.

Academician P. S. Alexandrov wrote:

If the development of today's mathematics undoubtedly proceeds under the sign of algebraization, the penetration of algebraic concepts and algebraic methods into the most diverse mathematical theories, then this became possible only after the works of Emmy Noether.

Einstein, in a note on her death, ranked Noether among the greatest creative geniuses in mathematics.

Scientific activity

Basically, Noether's works relate to algebra, where they contributed to the creation of a new direction, known as abstract algebra. Noether played a decisive role in this field (along with Emil Artin and her student B. L. van der Waerden). Hermann Weil wrote:

Much of what constitutes the content of the second volume of van der Waerden's Modern Algebra (now simply Algebra) must be Emmy Noether's.

The terms "Noetherian ring", "Noetherian module", normalization theorems and the Lasker-Noether ideal decomposition theorem are now fundamental.

Noether had a great influence on the algebraization of topology, showing that the so-called. the "Betty numbers" are just the ranks of the homology groups.

Noether made a great contribution to mathematical physics, where the fundamental theorem of theoretical physics (published in 1918) is named after her, linking conservation laws with system symmetries (for example, the homogeneity of time entails the law of conservation of energy). This fruitful approach is the basis of the famous series of books "Theoretical Physics" by Landau-Lifshitz. Noether's theorem is especially important in quantum field theory, where the conservation laws arising from the existence of a certain symmetry group are usually the main source of information about the properties of the objects under study.

Noether's ideas and scientific views had a huge impact on many mathematicians and physicists. She raised a number of students who became world-class scientists and continued the new directions discovered by Noether.

Mathematician Emmy Noether was a genius who initiated a new approach in physics

Noether's theorem is in theoretical physics what natural selection is in biology. If you were to write an equation that sums up everything we know about theoretical physics, it would have the names of Feynman, Schrödinger, Maxwell, and Dirac on one end. But if you write the name Noether on the other side of the equation, that would make up for them all.

Emmy Noether was born in Bavaria in 1882. She attended a boarding school and received a diploma giving the right to teach languages ​​- French and English. However, the girl soon realized that mathematics, which her father and brother studied at the University of Erlangen, interested her much more. Women were not allowed to enter higher educational institutions, but Emmy passed the entrance exam with an A plus and just attended lectures as a volunteer until the university began to accept girls for study. And Noether was able to get a Ph.D.

The girl began to engage in research work and, one might say, invented general algebra. This discipline studies algebraic systems (algebraic structures) and reduce them to the most abstract forms. Noether's goal was to understand how mathematical ideas correlate with each other and to build general mathematical structures. She never claimed to have discovered something revolutionary, but her work was a new approach in mathematics.

While Noether was writing her groundbreaking work at the University of Erlangen, she had neither a position nor a salary. The only thing she could do was to replace her father in mathematics lectures from time to time when he was sick.

Seven years later, mathematicians David Hilbert and Felix Klein invited Noether to work with them at the University of Göttingen. They wanted a woman to solve the problem of conservation of energy in Einstein's theory of general relativity. In an attempt to do this, Emmy formulated Noether's theorem, thus making one of the most significant contributions to theoretical physics.

Einstein spoke of the theorem as an example of "clear mathematical thinking". Moreover, the theorem has a simple formulation: each continuous symmetry of a physical system corresponds to a certain conservation law. By symmetry is meant that the physical process - or its mathematical description - remains the same when any aspect of the installation changes.

For example, an ideal pendulum that swings back and forth indefinitely is symmetrical in time. Based on Noether's theorem, everything that has time symmetry conserves energy. Thus, the pendulum does not lose energy. If the system has rotational symmetry - that is, it works the same way, regardless of orientation in space - then the angular momentum is conserved in it. This means that if the object is initially rotating, then it will continue to rotate indefinitely. The stability we see in the orbits of the planets is a consequence of symmetries that work together - the conservation of both energy and angular momentum of bodies.

Noether's theorem allows us to make deep connections between the results of experiments and the fundamental mathematical description of their physics. Thinking about physics in this case forms the basis of the type of theoretical leap that led physicists to theoretically predict the Higgs boson long before the particle could be detected by LHC research. Symmetry is so fundamental to physics that the Standard Model of particle physics is often named after its symmetry groups: U(1)×SU(2)×SU(3).

It is, of course, great that Noether made a radical revolution in physics - but at the same time she continued to work without pay, often lecturing for Hilbert and being his assistant. In 1922, 4 years after the publication of her theorem, the woman received the status of a freelance assistant professor, and they began to give her a small salary. Emmy lectured all over Europe.

When the Nazis came to power, Noether found herself out of a job because she was Jewish. She had to emigrate to America, where she became a visiting professor at the women's college in Bryn Mawr. In addition, Emmy Noether gave weekly lectures at Princeton. In Bryn Mawr, Noether first began working with women mathematicians. It is tragic that she was given only 2 years to enjoy it. Noether died in 1935 at the age of 53 after an unsuccessful operation to remove a cancerous tumor.

Many of the great physicists and mathematicians of the time, including Einstein, praised Emmy. In her era, pundits worked hard to keep women out of science. But Noether overcame this rule (possibly with the support of Einstein).

Even today in mathematics and physics, we can observe an asymmetry in the attitude towards female and male scientists (this is called the "Matilda effect in science"). As Noether said, once the symmetry is broken, something is lost.

Katie Mack
The woman who invented abstract algebra // Cosmos Magazine
Translation: Katyusha Shutova

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