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Hilbert's Seventh Problem
Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers (''Irrationalität und Transzendenz bestimmter Zahlen''). Statement of the problem Two specific equivalent questions are asked: #In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is then the ratio between base and side always transcendental? #Is a^b always transcendental, for algebraic a \not\in \ and irrational algebraic b? Solution The question (in the second form) was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by Theodor Schneider in 1935. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem. (The restriction to irrational ''b'' is important, since it is easy to see that a^b is algebraic for algebraic ''a'' and rational ''b''.) From the point of view of generalizations, this is the case :b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic. Life Early life and edu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfond–Schneider Theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. History It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider. Statement : If ''a'' and ''b'' are complex algebraic numbers with ''a'' ≠ 0, 1, and ''b'' not rational, then any value of ''ab'' is a transcendental number. Comments * The values of ''a'' and ''b'' are not restricted to real numbers; complex numbers are allowed (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational). * In general, is multivalued, where ln stands for the natural logarithm. This accounts for the phrase "any value of" in the theorem's statement. * An equivalent formulation of the theorem is the following: if ''α'' and ''γ'' are nonzero algebraic numbers, and we take any non-zero logarithm of ''α'', then is either rational or transcendental. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of Symposia In Pure Mathematics
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gelfond–Schneider Constant
The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two: :2 = ... which was proved to be a transcendental number by Rodion Kuzmin in 1930. In 1934, Aleksandr Gelfond and Theodor Schneider independently proved the more general ''Gelfond–Schneider theorem'', which solved the part of Hilbert's seventh problem described below. Properties The square root of the Gelfond–Schneider constant is the transcendental number :\sqrt=\sqrt^= .... This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either \sqrt^\sqrt is rational, which proves the theorem, or it is irrational (as it turns out to be), and then :\left(\sqrt^\right)^=\sqrt^=\sqrt^2=2 is an irrational to an irrational power that is rational, which proves the theorem. The proof is not constructive, as it does not say which of the two cases is true, but it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award honours the Canadian mathematician John Charles Fields. The Fields Medal is regarded as one of the highest honors a mathematician can receive, and has been described as the Nobel Prize of Mathematics, although there are several major differences, including frequency of award, number of awards, age limits, monetary value, and award criteria. According to the annual Academic Excellence Survey by ARWU, the Fields Medal is consistently regarded as the top award in the field of mathematics worldwide, and in another reputation survey conducted by IREG in 2013–14, the Fields Medal came closely after the Abel Prize as the second most prestigious international award in mathematics. The prize includes a monetary award which, since 2006, has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baker's Theorem
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier. Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1. History To simplify notation, let \mathbb be the set of logarithms to the base ''e'' of nonzero algebraic numbers, that is \mathbb = \left \, where \Complex denotes the set of complex numbers and \overline denotes the algebraic numbers (the algebraic completion of the rational numbers \Q). Using this notation, several results in transcendental number theory become much easier to state. For example the Hermite–L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Baker (mathematician)
Alan Baker (19 August 1939 – 4 February 2018) was an English mathematician, known for his work on effective methods in number theory, in particular those arising from transcendental number theory. Life Alan Baker was born in London on 19 August 1939. He attended Stratford Grammar School, East London, and his academic career started as a student of Harold Davenport, at University College London and later at Trinity College, Cambridge, where he received his PhD. He was a visiting scholar at the Institute for Advanced Study in 1970 when he was awarded the Fields Medal at the age of 31. In 1974 he was appointed Professor of Pure Mathematics at Cambridge University, a position he held until 2006 when he became an Emeritus. He was a fellow of Trinity College from 1964 until his death. His interests were in number theory, transcendence, logarithmic forms, effective methods, Diophantine geometry and Diophantine analysis. In 2012 he became a fellow of the American Mathematical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Form In Logarithms
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier. Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1. History To simplify notation, let \mathbb be the set of logarithms to the base ''e'' of nonzero algebraic numbers, that is \mathbb = \left \, where \Complex denotes the set of complex numbers and \overline denotes the algebraic numbers (the algebraic completion of the rational numbers \Q). Using this notation, several results in transcendental number theory become much easier to state. For example the Hermite–L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theodor Schneider
__NOTOC__ Theodor Schneider (7 May 1911, Frankfurt am Main – 31 October 1988, Freiburg im Breisgau) was a German mathematician, best known for providing proof of what is now known as the Gelfond–Schneider theorem. Schneider studied from 1929 to 34 in Frankfurt; he solved Hilbert's 7th problem in his PhD thesis, which then came to be known as the Gelfond–Schneider theorem. Later he became an assistant to Carl Ludwig Siegel in Göttingen, where he stayed until 1953. Then he became a professor in Erlangen (1953–59) and finally until his retirement in Freiburg (1959–1976). During his time in Freiburg he also served as the director of the Mathematical Research Institute of Oberwolfach from 1959 to 1963. His doctoral students include H. P. Schlickewei. Works *''Einführung in die Theorie der transzendenten Zahlen'', Springer 1957 (German, French translation 1959) *''Transzendenzuntersuchungen periodischer Funktionen'', Teil 1,2, Journal für die Reine und Angewandte Mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the ''Bulletin of the American Mathematical Society''. Earlier publications (in the original German) appeared in and Nature and influence of the problems Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer. For other problems, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Gelfond
Alexander Osipovich Gelfond (russian: Алекса́ндр О́сипович Ге́льфонд; 24 October 1906 – 7 November 1968) was a Soviet mathematician. Gelfond's theorem, also known as the Gelfond-Schneider theorem is named after him. Biography Alexander Gelfond was born in Saint Petersburg, Russian Empire, the son of a professional physician and amateur philosopher Osip Gelfond. He entered the Moscow State University in 1924, started his postgraduate studies there in 1927 and obtained his PhD in 1930. His advisors were Aleksandr Khinchin and Vyacheslav Stepanov. In 1930 he stayed for five months in Germany (in Berlin and Göttingen) where he worked with Edmund Landau, Carl Ludwig Siegel and David Hilbert. In 1931 he started teaching as a Professor at the Moscow State University and worked there until the last day of his life. Since 1933 he also worked at the Steklov Institute of Mathematics. In 1939 he was elected a Corresponding member of the Academy of Scienc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the polynomial . That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number 1 + i is algebraic because it is a root of . All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as and , are called transcendental numbers. The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental. Examples * All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer and a (non-zero) natural number , satisfies the above definition, because is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
