|
Absolute Galois Group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' that fix ''K''. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. (When ''K'' is a perfect field, ''K''sep is the same as an algebraic closure ''K''alg of ''K''. This holds e.g. for ''K'' of characteristic zero, or ''K'' a finite field.) Examples * The absolute Galois group of an algebraically closed field is trivial. * The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R, and its degree over R is ''C:Rnbsp;= 2. * The absolute Galois group of a finite field ''K'' is isomorphic to the group of profinite integers :: \hat = \varprojlim \mathbf/n\mathbf. :(For the notation, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate Picture
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * "Complex", a song by Katie Gregson-MacLeod, 2022 * "Complex" a song by Be'O and Zico, 2022 * Complex Networks, publisher of the now-only-online magazine ''Complex'' Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Profinite Integer
In mathematics, a profinite integer is an element of the ring (mathematics), ring (sometimes pronounced as zee-hat or zed-hat) :\widehat = \varprojlim \mathbb/n\mathbb, where the inverse limit of the quotient rings \mathbb/n\mathbb runs through all natural numbers n, Partial order, partially ordered by divisibility. By definition, this ring is the profinite completion of the integers \mathbb. By the Chinese remainder theorem, \widehat can also be understood as the direct product of rings :\widehat = \prod_p \mathbb_p, where the index p runs over all prime numbers, and \mathbb_p is the ring of p-adic integer, ''p''-adic integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of Ring of adeles, adeles. In addition, it provides a basic tractable example of a profinite group. Construction The profinite integers \widehat can be constructed as the set of sequences \upsilon of residues represented as \upsilon = (\upsilon_1 \bmod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is an algebraic number, because it is a root of the polynomial X^2 - X - 1, i.e., a solution of the equation x^2 - x - 1 = 0, and the complex number 1 + i is algebraic as a root of X^4 + 4. Algebraic numbers include all integers, rational numbers, and nth root, ''n''-th roots of integers. Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field (mathematics), field, denoted \overline. The set of algebraic real numbers \overline \cap \R is also a field. Numbers which are not algebraic are called transcendental number, transcendental and include pi, and . There are countable set, countably many algebraic numbers, hence almost all real (or complex) numbers (in the sense of Lebesgue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Real
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image (mathematics), image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one zero of a function, root of an integer polynomial ''P'', all of the roots of ''P'' being real; or that the tensor product of fields, tensor product algebra of ''F'' with the real field, over Q, is isomorphic to a tensor power of R. For example, quadratic fields ''F'' of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial ''P'' irreducible polynomial, irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will ''not'' be totally real, although it is a field (mathematics), fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory—indeed in any area where fields appear prominently. Definition and notation Suppose that ''E''/''F'' is a field extension. Then ''E'' may be considered as a vector space over ''F'' (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by 'E'':''F'' The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension ''E''/''F'' is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements). The degree should not be confused with the transcendence degree of a field; for example, the field Q(' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moshe Jarden
Moshe Jarden () is an Israeli mathematician, specialist in field arithmetic. Biography Moshe Jarden was born in 1942 in Tel Aviv. His father, Dr. Dov Jarden (1911–1986), was a mathematician, writer and linguist, who transmitted him his love to mathematics. In 1970 he received his Ph.D. in mathematics from the Hebrew University of Jerusalem, with Hillel Furstenberg as his thesis advisor. He accomplished his postdoctorate during 1971–1973 at the Institut of Mathematics, Heidelberg University, with Peter Roquette as his mentor, and habilitated there in 1972. During these years in Heidelberg, he initiated an intense and long-term cooperation with German mathematicians, especially with Peter Roquette, Wulf-Dieter Geyer, Gerhard Frey, and Juergen Ritter. His achievements in mathematics, as well as the foundation of this fruitful cooperation with German mathematicians, earned him the L. Meithner-A.v.Humboldt Prize by the Alexander von Humboldt Foundation in 2001. In 1974, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dan Haran
Dan or DAN may refer to: People * Dan (name), including a list of people with the name ** Dan (king), several kings of Denmark * Dan people, an ethnic group located in West Africa **Dan language, a Mande language spoken primarily in Côte d'Ivoire and Liberia * Dan (son of Jacob), one of the 12 sons of Jacob/Israel in the Bible **Tribe of Dan, one of the 12 tribes of Israel descended from Dan **Danel, the hero figure of Ugarit who inspired stories of the biblical figure * Crown Prince Dan, prince of Yan in ancient China Places * Dan (ancient city), the biblical location also called Dan, and identified with Tel Dan * Dan, Israel, a kibbutz * Dan, subdistrict of Kap Choeng District, Thailand * Dan, West Virginia, an unincorporated community in the United States * Dan River (other) * Danzhou, formerly Dan County, China * Gush Dan, the metropolitan area of Tel Aviv in Israel Organizations *Dan-Air, a defunct airline in the United Kingdom *Dan Bus Company, a public transpo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Florian Pop
Florian Pop (born 1952 in Zalău) is a Romanian mathematician, a professor of mathematics at the University of Pennsylvania. Pop received his Ph.D. in 1987 and his habilitation in 1991, both from the University of Heidelberg. He has been a member of the Institute for Advanced Study in Princeton, and (from 1996 to 2003) a professor at the University of Bonn prior to joining the University of Pennsylvania faculty. Pop's research concerns algebraic geometry, arithmetic geometry, anabelian geometry, and Galois theory. call his habilitation thesis, concerning the characterization of certain fields by their absolute Galois groups, a "milestone". In 1996, Pop was awarded the Gay-Lussac–von Humboldt Prize for Mathematics, and in 2003 he was awarded the Romanian Order of Merit, Commander rank. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Harbater
David Harbater (born December 19, 1952) is an American mathematician at the University of Pennsylvania, well known for his work in Galois theory, algebraic geometry and arithmetic geometry. Early life and education Harbater was born in New York City and attended Stuyvesant High School, where he was on the math team. After graduating in 1970, he entered Harvard University. After graduating summa cum laude in 1974, Harbater earned a master's degree from Brandeis University and then a Ph.D. in 1978 from MIT, where he wrote a dissertation (Deformation Theory and the Fundamental Group in Algebraic Geometry) under the direction of Michael Artin. Research He solved the inverse Galois problem over \mathbb_p(t), and made many other significant contributions to the field of Galois theory. Harbater's recent work on patching over fields, together with Julia Hartmann and Daniel Krashen, has had applications in such varied fields as quadratic forms, central simple algebras and local-global p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann's Existence Theorem
In mathematics, specifically complex analysis, Riemann's existence theorem says, in modern formulation, that the category of compact Riemann surfaces is equivalent to the category of complex complete algebraic curves. Sometimes, the theorem also refers to a generalization (a theorem of Grauert–Remmert), which says that the category of finite topological coverings of a complex algebraic variety is equivalent to the category of finite étale coverings of the variety. Original statement Let ''X'' be a compact Riemann surface, p_1, \cdots, p_s distinct points in ''X'' and a_1, \cdots, a_s complex numbers. Then there is a meromorphic function f on ''X'' such that f(p_i) = a_i for each ''i''. Proof For now, see SGA 1, Expose XII, Théorème 5.1., or SGA 4, Expose XI. 4.3. Consequences There are a number of consequences. By definition, if ''X'' is a complex algebraic variety, the étale fundamental group of ''X'' at a geometric point ''x'' is the projective limit :\pi_1^(X, x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adrien Douady
Adrien Douady (; 25 September 1935 – 2 November 2006) was a French mathematician born in La Tronche, Isère. He was the son of Daniel Douady and Guilhen Douady. Douady was a student of Henri Cartan at the École normale supérieure, and initially worked in homological algebra. His thesis concerned deformations of complex analytic spaces. Subsequently, he became more interested in the work of Pierre Fatou and Gaston Julia and made significant contributions to the fields of analytic geometry and dynamical systems. Together with his former student John H. Hubbard, he launched a new subject, and a new school, studying properties of iterated quadratic complex mappings. They made important mathematical contributions in this field of complex dynamics, including a study of the Mandelbrot set. One of their most fundamental results is that the Mandelbrot set is connected; perhaps most important is their theory of renormalization of (polynomial-like) maps. The Douady rabbit, a qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
