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Igor Shafarevich
Igor Rostislavovich Shafarevich (; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised socialism and other books which were described as anti-semitic. Mathematics From his early years, Shafarevich made fundamental contributions to several parts of mathematics including algebraic number theory, algebraic geometry and arithmetic algebraic geometry. In particular, in algebraic number theory, the Shafarevich–Weil theorem extends the commutative reciprocity map to the case of Galois groups, which are central extensions of abelian groups by finite groups. Shafarevich was the first mathematician to give a completely self-contained formula for the Hilbert pairing, thus initiating an important branch of the study of explicit formulas in number theory. Another famous (and slightly incomplete) result is Shafarevich's theorem on solvable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zhytomyr
Zhytomyr ( ; see #Names, below for other names) is a city in the north of the western half of Ukraine. It is the Capital city, administrative center of Zhytomyr Oblast (Oblast, province), as well as the administrative center of the surrounding Zhytomyr urban hromada (hromada, commune) and Zhytomyr Raion (Raion, district). Moreover Zhytomyr consists of two Urban districts of Ukraine, urban districts: Bohunskyi District and Koroliovskyi District (named in honour of Sergey Korolyov). Zhytomyr occupies an area of . Its population is Zhytomyr is a major transport hub. The city lies on a historic route linking the city of Kyiv with the west through Brest, Belarus, Brest. Today it links Warsaw with Kyiv, Minsk with Izmail, and several major cities of Ukraine. Zhytomyr was also the location of Ozerne (air base), Ozerne airbase, a key Cold War strategic aircraft base southeast of the city. Important economic activities of Zhytomyr include lumber milling, food processing, granite quarr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golod–Shafarevich Theorem
In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite. The inequality Let ''A'' = ''K''⟨''x''1, ..., ''x''''n''⟩ be the free algebra over a field ''K'' in ''n'' = ''d'' + 1 non-commuting variables ''x''''i''. Let ''J'' be the 2-sided ideal of ''A'' generated by homogeneous elements ''f''''j'' of ''A'' of degree ''d''''j'' with :2 ≤ ''d''1 ≤ ''d''2 ≤ ... where ''d''''j'' tends to infinity. Let ''r''''i'' be the number of ''d''''j'' equal to ''i''. Let ''B''=''A''/''J'', a graded algebra. Let ''b''''j'' = dim ''B''''j''. The ''fundamental inequality'' of Golod and Shafarevich states that :: b_j\ge nb_ -\sum_^ b_ r_i. As a consequence: * ''B'' is infinite-dimensional if ''r''''i'' ≤ ''d''2/4 for all ''i'' Applications This result has important applica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as E/F and read "''E'' over ''F''). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism \alpha:E\to E such that \alpha(x) = x for each x\in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by \operatorname(E/F). If E/F is a Galois extension, then \op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Algebraic Geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers. Overview The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antisemitism
Antisemitism or Jew-hatred is hostility to, prejudice towards, or discrimination against Jews. A person who harbours it is called an antisemite. Whether antisemitism is considered a form of racism depends on the school of thought. Antisemitic tendencies may be motivated primarily by negative sentiment towards Jewish peoplehood, Jews as a people or negative sentiment towards Jews with regard to Judaism. In the former case, usually known as racial antisemitism, a person's hostility is driven by the belief that Jews constitute a distinct race with inherent traits or characteristics that are repulsive or inferior to the preferred traits or characteristics within that person's society. In the latter case, known as religious antisemitism, a person's hostility is driven by their religion's perception of Jews and Judaism, typically encompassing doctrines of supersession that expect or demand Jews to turn away from Judaism and submit to the religion presenting itself as Judaism's suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Socialism
Socialism is an economic ideology, economic and political philosophy encompassing diverse Economic system, economic and social systems characterised by social ownership of the means of production, as opposed to private ownership. It describes the Economic ideology, economic, Political philosophy, political, and Social theory, social theories and Political movement, movements associated with the implementation of such systems. Social ownership can take various forms, including State ownership, public, Community ownership, community, Collective ownership, collective, cooperative, or Employee stock ownership, employee.: "Just as private ownership defines capitalism, social ownership defines socialism. The essential characteristic of socialism in theory is that it destroys social hierarchies, and therefore leads to a politically and economically egalitarian society. Two closely related consequences follow. First, every individual is entitled to an equal ownership share that earns an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russian Federation
Russia, or the Russian Federation, is a country spanning Eastern Europe and North Asia. It is the list of countries and dependencies by area, largest country in the world, and extends across Time in Russia, eleven time zones, sharing Borders of Russia, land borders with fourteen countries. Russia is the List of European countries by population, most populous country in Europe and the List of countries and dependencies by population, ninth-most populous country in the world. It is a Urbanization by sovereign state, highly urbanised country, with sixteen of its urban areas having more than 1 million inhabitants. Moscow, the List of metropolitan areas in Europe, most populous metropolitan area in Europe, is the capital and List of cities and towns in Russia by population, largest city of Russia, while Saint Petersburg is its second-largest city and Society and culture in Saint Petersburg, cultural centre. Human settlement on the territory of modern Russia dates back to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
