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Hyperfinite Field
In mathematics, a hyper-finite field is an uncountable field similar in many ways to finite fields. More precisely a field ''F'' is called hyper-finite if it is uncountable and quasi-finite, and for every subfield ''E'', every absolutely entire ''E''-algebra ( regular field extension of ''E'') of smaller cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ... than ''F'' can be embedded in ''F''. They were introduced by . Every hyper-finite field is a pseudo-finite field, and is in particular a model for the first-order theory of finite fields. References * Field (mathematics) {{Abstract-algebra-stub ... [...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] |
Uncountable Set
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the natural numbers. Examples of uncountable sets include the set of all real numbers and set of all subsets of the natural numbers. Characterizations There are many equivalent characterizations of uncountability. A set ''X'' is uncountable if and only if any of the following conditions hold: * There is no injective function (hence no bijection) from ''X'' to the set of natural numbers. * ''X'' is nonempty and for every ω- sequence of elements of ''X'', there exists at least one element of X not included in it. That is, ''X'' is nonempty and there is no surjective function from the natural numbers to ''X''. * The cardinality of ''X'' is neither finite nor equal to \aleph_0 ( aleph-null). * The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-finite Field
In mathematics, a quasi-finite field is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is ''finite'' (i.e. non-archimedean local fields), but the theory applies equally well when the residue field is only assumed quasi-finite. Formal definition A quasi-finite field is a perfect field ''K'' together with an isomorphism of topological groups : \phi : \hat \to \operatorname(K_s/K), where ''K''''s'' is an algebraic closure of ''K'' (necessarily separable because ''K'' is perfect). The field extension ''K''''s''/''K'' is infinite, and the Galois group is accordingly given the Krull topology. The group \widehat is the profinite completion of integers with respect to its subgroups of finite index. This definition is equivalent to saying that ''K'' has a unique (necessarily cyclic) extension ''K''''n'' of degree ''n'' for each integer ''n'' ≥ 1, and that the union of these extensions is equal to ''K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Field Extension
In field theory, a branch of algebra, a field extension L/k is said to be regular if ''k'' is algebraically closed in ''L'' (i.e., k = \hat k where \hat k is the set of elements in ''L'' algebraic over ''k'') and ''L'' is separable over ''k'', or equivalently, L \otimes_k \overline is an integral domain when \overline is the algebraic closure of k (that is, to say, L, \overline are linearly disjoint over ''k'').Fried & Jarden (2008) p.38Cohn (2003) p.425 Properties * Regularity is transitive: if ''F''/''E'' and ''E''/''K'' are regular then so is ''F''/''K''.Fried & Jarden (2008) p.39 * If ''F''/''K'' is regular then so is ''E''/''K'' for any ''E'' between ''F'' and ''K''. * The extension ''L''/''k'' is regular if and only if every subfield of ''L'' finitely generated over ''k'' is regular over ''k''. * Any extension of an algebraically closed field is regular.Cohn (2003) p.426 * An extension is regular if and only if it is separable and primary.Fried & Jarden (2008) p.44 * A purel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thumb is ''pollex'' (compare ''hallux'' for big toe), and the corresponding adjective for thumb is ''pollical''. Definition Thumb and fingers The English word ''finger'' has two senses, even in the context of appendages of a single typical human hand: 1) Any of the five terminal members of the hand. 2) Any of the four terminal members of the hand, other than the thumb. Linguistically, it appears that the original sense was the first of these two: (also rendered as ) was, in the inferred Proto-Indo-European language, a suffixed form of (or ), which has given rise to many Indo-European-family words (tens of them defined in English dictionaries) that involve, or stem from, concepts of fiveness. The thumb shares the following with each of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-finite Field
In mathematics, a pseudo-finite field ''F'' is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that ''F'' is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely irreducible variety over ''F'' has a point defined over ''F''). Every hyperfinite field is pseudo-finite and every pseudo-finite field is quasifinite. Every non-principal ultraproduct of finite fields is pseudo-finite. Pseudo-finite fields were introduced by . References * * {{citation , last1=Fried , first1=Michael D. , last2=Jarden , first2=Moshe , title=Field arithmetic , edition=3rd revised , series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge , volume=11 , publisher=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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
