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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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original paginatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schanuel's Conjecture
In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers. Statement The conjecture is as follows: :Given any complex numbers that are linearly independent over the rational numbers \mathbb, the field extension \mathbb(''z''1, ..., ''z''''n'', ''e''''z''1, ..., ''e''''z''''n'') has transcendence degree at least over \mathbb. The conjecture can be found in Lang (1966). Consequences The conjecture, if proven, would generalize most known results in transcendental number theory. The special case where the numbers ''z''1,...,''z''''n'' are all algebraic is the Lindemann–Weierstrass theorem. If, on the other hand, the numbers are chosen so as to make exp(''z''1),...,exp(''z''''n'') all algebraic then one would prove that linearly independent logarithms of algebraic numbers are algebraically independent, a st ... [...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 Hermiteâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindemann–Weierstrass Theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transcendence degree over \mathbb. An equivalent formulation , is the following: This equivalence transforms a linear relation over the algebraic numbers into an algebraic relation over \mathbb by using the fact that a symmetric polynomial whose arguments are all conjugates of one another gives a rational number. The theorem is named for Ferdinand von Lindemann and Karl Weierstrass. Lindemann proved in 1882 that is transcendental for every non-zero algebraic number thereby establishing that is transcendental (see below). Weierstrass proved the above more general statement in 1885. The theorem, along with the Gelfond–Schneider theorem, is extended by Baker's theorem, and all of these would be further generalized by Schanuel's conject ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Gelfond's Constant
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is , that is, raised to the power . Like both and , this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that e^\pi = (e^)^ = (-1)^, where is the imaginary unit. Since is algebraic but not rational, is transcendental. The constant was mentioned in Hilbert's seventh problem. A related constant is , known as the Gelfond–Schneider constant. The related value + is also irrational. Numerical value The decimal expansion of Gelfond's constant begins :e^\pi = ... Construction If one defines and k_ = \frac for , then the sequence (4/k_)^ converges rapidly to . Continued fraction expansion e^ = 23+ \cfrac This is based on the digits for the simple continued fraction: e^ = [23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, 108, 2, 2, 1, 3, 1, ... [...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] |
P-adic Exponential Function
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
