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Pseudo Algebraically Closed
In mathematics, a field K is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.Fried & Jarden (2008) p.218 Formulation A field ''K'' is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds: *Each absolutely irreducible variety V defined over K has a K-rational point. *For each absolutely irreducible polynomial f\in K _1,T_2,\cdots ,T_r,X/math> with \frac\not =0 and for each nonzero g\in K _1,T_2,\cdots ,T_r/math> there exists (\textbf,b)\in K^ such that f(\textbf,b)=0 and g(\textbf)\not =0. *Each absolutely irreducible polynomial f\in K ,X/math> has infinitely many K-rational points. *If R is a finitely generated integral domain over K with quotient field which is regular over K, then there exist a homomorphism h:R\to K such that h(a) = a for each a \in K. Examples * Algebraically closed fields and separably closed fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Severi–Brauer Variety
In mathematics, a Severi–Brauer variety over a field (mathematics), field ''K'' is an algebraic variety ''V'' which becomes isomorphic to a projective space over an algebraic closure of ''K''. The varieties are associated to central simple algebras in such a way that the algebra splits over ''K'' if and only if the variety has a rational point over ''K''. studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group. In dimension one, the Severi–Brauer varieties are conic section, conics. The corresponding central simple algebras are the quaternion algebras. The algebra corresponds to the conic with equation : z^2 = ax^2 + by^2 \ and the algebra ''splits'', that is, is isomorphic to a Matrix ring, matrix algebra over ''K'', if and only if has a point defined over ''K'': this is in turn equivalent to being isomorphic to the projective line over ''K''. Such varieties are of interest not only in diophantine ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer Group
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer. The Brauer group arose out of attempts to classify division algebras over a field. It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles. Construction A central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' such that ''A'' is a simple ring and the center of ''A'' is equal to ''K''. Note that CSAs are in general ''not'' division algebras, though CSAs can be used to classify division algebras. For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ''F''. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If ''E'' is a given field, and ''G'' is a finite group of automorphisms of ''E'' with fixed field ''F'', then ''E''/''F'' is a Galois extension. The property of an extension being Galois behaves well with respect to field composition and intersection. Characterization of Galois extensions An important theorem of Emil Artin states that for a finite extension E/F, each of the following statements is equivalent to the statement that E/F is Galois: *E/F is a normal extension and a separable extension. *E is a splitting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Real Field
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polynomial ''P'', all of the roots of ''P'' being real; or that the 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 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 of real numbers. The totally real number fields play a significant special role in algebraic number theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In an algebraic structure such as a group, a ring, or vector space, an ''automorphism'' is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism f: X\to X is an automorphism if there is a morphism g: X\to X such that g\circ f= f\circ g = \operatorname _X, where \operatorname _X is the identity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗. More precisely, is a subgroup of if the Restriction (mathematics), restriction of ∗ to is a group operation on . This is often denoted , read as " is a subgroup of ". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group is a subgroup which is a subset, proper subset of (that is, ). This is often represented notationally by , read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is, ). If is a subgroup of , then is sometimes called an overgroup of . The same definitions apply more generally when is an arbitrary se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbertian Field
In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within ''K'' a polynomial that does not always factorise. One is also allowed to take finite unions. Formulation More precisely, let ''V'' be an algebraic variety over ''K'' (assumptions here are: ''V'' is an irreducible set, a quasi-projective variety, and ''K'' has characteristic zero). A type I thin set is a subset of ''V''(''K'') that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than ''d'', the dimension of ''V''. A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the ''K''-points of some other ''d''-dimensional a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
