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Evgeny Golod
Evgenii Solomonovich Golod (russian: Евгений Соломонович Голод, 21 October 1935 – 5 July 2018) was a Russian mathematician who proved the Golod–Shafarevich theorem on class field tower A tower is a tall structure, taller than it is wide, often by a significant factor. Towers are distinguished from masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting structures. Towers are specifi ...s. As an application, he gave a negative solution to the Kurosh–Levitzky problem on the nilpotency of finitely generated nil algebras, and so to a weak form of Burnside's problem. Golod was a student of Igor Shafarevich. As of 2015, Golod had 39 academic descendants, most of them through his student Luchezar L. Avramov. Selected publications * * References {{DEFAULTSORT:Golod, Evgeny S. 1935 births 2018 deaths Russian mathematicians Scientists from Moscow ... [...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 applicatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Class Field
In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' over ''K'' is canonically isomorphic to the ideal class group of ''K'' using Frobenius elements for prime ideals in ''K''. In this context, the Hilbert class field of ''K'' is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of ''K''. That is, every real embedding of ''K'' extends to a real embedding of ''E'' (rather than to a complex embedding of ''E''). Examples *If the ring of integers of ''K'' is a unique factorization domain, in particular if K = \mathbb , then ''K'' is its own Hilbert class field. *Let K = \mathbb(\sqrt) of discriminant -15. The field L = \mathbb(\sqrt, \sqrt) has discriminant 225=(-15)^2 and so is an everywhere unramified extension of ''K'', and it is abelian. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tower Of Fields
In mathematics, a tower of fields is a sequence of field extensions : The name comes from such sequences often being written in the form :\begin\vdots \\ , \\ F_2 \\ , \\ F_1 \\ , \\ \ F_0. \end A tower of fields may be finite or infinite. Examples * is a finite tower with rational, real and complex numbers. *The sequence obtained by letting ''F''0 be the rational numbers Q, and letting ::F_ = F_n\!\left(2^\right) :(i.e. ''F''''n''+1 is obtained from ''F''''n'' by adjoining a 2''n'' th root of 2) is an infinite tower. *If ''p'' is a prime number the ''p'' th cyclotomic tower of Q is obtained by letting ''F''0 = Q and ''F''''n'' be the field obtained by adjoining to Q the ''pn'' th roots of unity. This tower is of fundamental importance in Iwasawa theory. *The Golod–Shafarevich theorem shows that there are infinite towers obtained by iterating the Hilbert class field construction to a number field. References *Section 4.1.4 of {{Citation , last ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burnside's Problem
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants (see bounded and restricted below) that differ in the additional conditions imposed on the orders of the group elements, some of which are still open questions. Brief history Initial work pointed towards the affirmative answer. For example, if a group ''G'' is finitely generated and the order of each element of ''G'' is a divisor of 4, then ''G'' is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Igor Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 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 (controversially) 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 theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Genealogy
An academic, or scientific genealogy organizes a family tree of scientists and scholars according to mentoring relationships, often in the form of dissertation supervision relationships, and not according to genetic relationships as in conventional genealogy. Since the term ''academic genealogy'' has now developed this specific meaning, its additional use to describe a more academic approach to conventional genealogy would be ambiguous, so the description scholarly genealogy is now generally used in the latter context. Overview The academic lineage or academic ancestry of someone is a chain of professors who have served as academic mentors or thesis advisors of each other, ending with the person in question. Many genealogical terms are often recast in terms of academic lineages, so one may speak of academic descendants, children, siblings, etc. One method of developing an academic genealogy is to organize individuals by prioritizing their degree of relationship to a mentor/ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 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 (controversially) 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 numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1935 Births
Events January * January 7 – Italian premier Benito Mussolini and French Foreign Minister Pierre Laval conclude Franco-Italian Agreement of 1935, an agreement, in which each power agrees not to oppose the other's colonial claims. * January 12 – Amelia Earhart becomes the first person to successfully complete a solo flight from Hawaii to California, a distance of 2,408 miles. * January 13 – A plebiscite in the Saar (League of Nations), Territory of the Saar Basin shows that 90.3% of those voting wish to join Germany. * January 24 – The first canned beer is sold in Richmond, Virginia, United States, by Gottfried Krueger Brewing Company. February * February 6 – Parker Brothers begins selling the board game Monopoly (game), Monopoly in the United States. * February 13 – Richard Hauptmann is convicted and sentenced to death for the kidnapping and murder of Charles Lindbergh Jr. in the United States. * February 15 – The discovery and clinical development of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2018 Deaths
This is a list of deaths of notable people, organised by year. New deaths articles are added to their respective month (e.g., Deaths in ) and then linked here. 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 1988 1987 See also * Lists of deaths by day The following pages, corresponding to the Gregorian calendar, list the historical events, births, deaths, and holidays and observances of the specified day of the year: Footnotes See also * Leap year * List of calendars * List of non-standard ... * Deaths by year {{DEFAULTSORT:deaths by year ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Russian Mathematicians
Russian(s) refers to anything related to Russia, including: *Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *Rossiyane (), Russian language term for all citizens and people of Russia, regardless of ethnicity *Russophone, Russian-speaking person (, ''russkogovoryashchy'', ''russkoyazychny'') *Russian language, the most widely spoken of the Slavic languages *Russian alphabet *Russian cuisine *Russian culture *Russian studies Russian may also refer to: *Russian dressing *''The Russians'', a book by Hedrick Smith *Russian (comics), fictional Marvel Comics supervillain from ''The Punisher'' series *Russian (solitaire), a card game * "Russians" (song), from the album ''The Dream of the Blue Turtles'' by Sting *"Russian", from the album ''Tubular Bells 2003'' by Mike Oldfield *"Russian", from the album '' '' by Caravan Palace *Nik Russian, the perpetrator of a con committed in 2002 *The South African name for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
