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75 relations: Alan Baker (mathematician), Alexander Gelfond, Algebraic equation, Algebraic function, Algebraic independence, Algebraic number, Almost all, American Mathematical Monthly, Apéry's constant, Auxiliary function, Axel Thue, Baker's theorem, Bijection, Boris Zilber, Cambridge University Press, Cantor's diagonal argument, Carl Ludwig Siegel, Champernowne constant, Charles Hermite, Class number problem, Closed-form expression, Coefficient, Compass-and-straightedge construction, Complex number, Constant problem, Continued fraction, Crelle's Journal, David Hilbert, Degree of a polynomial, Diophantine approximation, Diophantine equation, Euler's identity, Euler–Mascheroni constant, Ferdinand von Lindemann, Fields Medal, Function (mathematics), Fundamental theorem of algebra, Gelfond's constant, Gelfond–Schneider constant, Gelfond–Schneider theorem, Georg Cantor, Georg Cantor's first set theory article, Gisbert Wüstholz, Gottfried Wilhelm Leibniz, Hilbert's problems, Hilbert's seventh problem, Integer, Joseph Liouville, Karl Weierstrass, Klaus Roth, ..., Leonhard Euler, Lindemann–Weierstrass theorem, Linear independence, Liouville number, Mathematische Annalen, Mathematische Zeitschrift, Men of Mathematics, Model theory, Natural number, Necessity and sufficiency, Number theory, Polynomial, Rational number, Roth's theorem, Schanuel's conjecture, Serge Lang, Set theory, Siegel's lemma, Sine, Squaring the circle, Stephen Schanuel, Theodor Schneider, Transcendence degree, Transcendental number, Uncountable set. Expand index (25 more) »
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.
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Alexander Gelfond
Alexander Osipovich Gelfond (Алекса́ндр О́сипович Ге́льфонд; 24 October 1906 – 7 November 1968) was a Soviet mathematician.
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Algebraic equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.
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Algebraic function
In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation.
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Algebraic independence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set is algebraically independent over K if and only if α is transcendental over K. In general, all the elements of an algebraically independent set S over K are by necessity transcendental over K, and over all of the field extensions over K generated by the remaining elements of S.
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Algebraic number
An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).
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Almost all
In mathematics, the term "almost all" means "all but a negligible amount".
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American Mathematical Monthly
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.
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Apéry's constant
In mathematics, at the intersection of number theory and special functions, Apéry's constant is defined as the number where is the Riemann zeta function.
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Auxiliary function
In mathematics, auxiliary functions are an important construction in transcendental number theory.
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Axel Thue
Axel Thue (19 February 1863 – 7 March 1922), was a Norwegian mathematician, known for highly original work in diophantine approximation, and combinatorics.
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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.
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Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
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Boris Zilber
Boris Zilber (Борис Иосифович Зильбер, born 1949) is a Soviet-British mathematician who works in mathematical logic (specifically model theory).
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Cantor's diagonal argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.
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Carl Ludwig Siegel
Carl Ludwig Siegel (December 31, 1896 – April 4, 1981) was a German mathematician specialising in number theory and celestial mechanics.
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Champernowne constant
In mathematics, the Champernowne constant is a transcendental real constant whose decimal expansion has important properties.
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Charles Hermite
Prof Charles Hermite FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
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Class number problem
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers d) having class number n. It is named after Carl Friedrich Gauss.
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Closed-form expression
In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.
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Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.
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Compass-and-straightedge construction
Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Constant problem
In mathematics, the constant problem is the problem of deciding if a given expression is equal to zero.
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Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.
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Crelle's Journal
Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik (in English: Journal for Pure and Applied Mathematics).
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David Hilbert
David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.
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Degree of a polynomial
The degree of a polynomial is the highest degree of its monomials (individual terms) with non-zero coefficients.
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Diophantine approximation
In number theory, the field of Diophantine approximation deals with the approximation of real numbers by rational numbers.
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Diophantine equation
In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values).
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Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality where Euler's identity is named after the Swiss mathematician Leonhard Euler.
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Euler–Mascheroni constant
The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.
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Ferdinand von Lindemann
Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that pi (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficients.
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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.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
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Gelfond's constant
In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is e, that is, e raised to the power pi.
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Gelfond–Schneider constant
The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two: which was proved to be a transcendental number by Rodion Kuzmin in 1930.
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Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.
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Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.
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Georg Cantor's first set theory article
Georg Cantor's first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties.
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Gisbert Wüstholz
Gisbert Wüstholz is a German mathematician who obtained his Ph.D. from the Albert Ludwigs University of Freiburg in 1977, under the supervision of Theodor Schneider.
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Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz (or; Leibnitz; – 14 November 1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy.
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Hilbert's problems
Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.
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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.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Joseph Liouville
Joseph Liouville FRS FRSE FAS (24 March 1809 – 8 September 1882) was a French mathematician.
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Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".
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Klaus Roth
Klaus Friedrich Roth (29 October 1925 – 10 November 2015) was a German-born British mathematician known for work on diophantine approximation, the large sieve, and irregularities of distribution.
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Leonhard Euler
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
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Lindemann–Weierstrass theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers.
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Linear independence
In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.
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Liouville number
In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers.
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Mathematische Annalen
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.
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Mathematische Zeitschrift
Mathematische Zeitschrift (German for Mathematical Journal) is a mathematical journal for pure and applied mathematics published by Springer Verlag.
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Men of Mathematics
Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré is a book on the history of mathematics published in 1937 by Scottish-born American mathematician and science fiction writer E. T. Bell (1883–1960).
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Model theory
In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.
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Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
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Necessity and sufficiency
In logic, necessity and sufficiency are terms used to describe an implicational relationship between statements.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Roth's theorem
In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers.
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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.
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Serge Lang
Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician and activist.
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Set theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
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Siegel's lemma
In transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions.
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Sine
In mathematics, the sine is a trigonometric function of an angle.
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Squaring the circle
Squaring the circle is a problem proposed by ancient geometers.
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Stephen Schanuel
Stephen H. Schanuel (1934—2014) was an American mathematician working in the fields of abstract algebra and category theory, number theory, and measure theory.
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Theodor Schneider
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.
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Transcendence degree
In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension.
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Transcendental number
In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.
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Uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.
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Redirects here:
Transcendence measure, Transcendence theory.
References
[1] https://en.wikipedia.org/wiki/Transcendental_number_theory
