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P-adically Closed Field
In mathematics, a ''p''-adically closed field is a field that enjoys a closure property that is a close analogue for ''p''-adic fields to what real closure is to the real field. They were introduced by James Ax and Simon B. Kochen in 1965. Definition Let K be the field \mathbb of rational numbers and v be its usual p-adic valuation (with v(p)=1). If F is a (not necessarily algebraic) extension field of K, itself equipped with a valuation w, we say that (F,w) is formally ''p''-adic when the following conditions are satisfied: * w extends v (that is, w(x)=v(x) for all x\in K), * the residue field of w coincides with the residue field of v (the residue field being the quotient of the valuation ring \ by its maximal ideal \), * the smallest positive value of w coincides with the smallest positive value of v (namely 1, since ''v'' was assumed to be normalized): in other words, a uniformizer for K remains a uniformizer for F. (Note that the value group of ''K'' may be larger tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Maximal Ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals contained between ''I'' and ''R''. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. The set of maximal ideals of a unital commutative ring ''R'', typically equipped with the Zariski topology, is known as the maximal spectrum of ''R'' and is variously denoted m-Spec ''R'', Specm ''R'', MaxSpec ''R'', or Spm ''R''. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal ideal ''A'' is not necessarily ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quantifier Elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that ..." can be viewed as a question "When is there an x such that ...?", and the statement without quantifiers can be viewed as the answer to that question. One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest. A theory has quantifier elimination if for every formula \alpha, there exists another formula \alpha_ without quantifiers that is equivalent to it (modulo this theory). Examples An example from mathematics says that a single-variable quadratic polynomial has a real root if and only if its discriminant is non-negative: \exists x\in\mathbb. (a\neq 0 \wedge ax^2+bx+c=0)\ \ \Longleftrightarrow\ \ a\neq 0 \wedge b^2-4ac\geq 0 Here the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Model Complete Theory
In model theory, a first-order logic, first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson. Model companion and model completion A companion of a theory ''T'' is a theory ''T''* such that every model of ''T'' can be embedded in a model of ''T''* and vice versa. A model companion of a theory ''T'' is a companion of ''T'' that is model complete. Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if ''T'' is an \aleph_0-categorical theory, then it always has a model companion. A model completion for a theory ''T'' is a model companion ''T''* such that for any model ''M'' of ''T'', the theory of ''T''* together with the Diagram (mathematical logic), diagram of ''M'' is complete theory, complete. Roughly speaking, this means ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complete Theory
In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence \varphi, the theory T contains the sentence or its negation but not both (that is, either T \vdash \varphi or T \vdash \neg \varphi). Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem. This sense of ''complete'' is distinct from the notion of a complete ''logic'', which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness. Complete theories are closed under a number of conditions internally modelling the T-schema: * For a set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lexicographical Order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements. Another variant, widely used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences, to which the lexicographical order is applied. A generalization defines an order on an ''n''-ary Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered. Definition The words in a lexicon (the set of words used in some language) have a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
The Kochen Operator
''The'' is a grammatical article in English, denoting nouns that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with nouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of the archaic pronoun ''thee'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Henselian Ring
In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now restrict them to be commutative. Some standard references for Hensel rings are , , and . Definitions In this article rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings. * A local ring ''R'' with maximal ideal ''m'' is called Henselian if Hensel's lemma holds. This means that if ''P'' is a monic polynomial in ''R'' 'x'' then any factorization of its image ''P'' in (''R''/''m'') 'x''into a product of coprime monic polynomials can be lifted to a factorization in ''R'' 'x'' * A local ring is Henselian if and only if every finite ring extension is a product of local rings. * A Henselian local ring is called strictly Henselian if its residue field is separably closed. * By abuse of term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gaussian Rational
In mathematics, a Gaussian rational number is a complex number of the form ''p'' + ''qi'', where ''p'' and ''q'' are both rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(''i''), obtained by adjoining the imaginary number ''i'' to the field of rationals Q. Properties of the field The field of Gaussian rationals provides an example of an algebraic number field that is both a quadratic field and a cyclotomic field (since ''i'' is a 4th root of unity). Like all quadratic fields it is a Galois extension of Q with Galois group cyclic of order two, in this case generated by complex conjugation, and is thus an abelian extension of Q, with conductor 4. As with cyclotomic fields more generally, the field of Gaussian rationals is neither ordered nor complete (as a metric space). The Gaussian integers Z 'i''form the ring of integers of Q(''i''). The set of all Gaussian rationals is countably infinite. The field of Gaussian ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Uniformizer
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' that satisfies any and all of the following equivalent conditions: # ''R'' is a local ring, a principal ideal domain, and not a field. # ''R'' is a valuation ring with a value group isomorphic to the integers under addition. # ''R'' is a local ring, a Dedekind domain, and not a field. # ''R'' is Noetherian and a local domain whose unique maximal ideal is principal, and not a field. # ''R'' is integrally closed, Noetherian, and a local ring with Krull dimension one. # ''R'' is a principal ideal domain with a unique non-zero prime ideal. # ''R'' is a principal ideal domain with a unique irreducible element (up to multiplication by units). # ''R'' is a unique factorization domain with a unique irreducible element (up to multiplication by units). # ''R'' is Noetherian, not a field, and every nonzero fraction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Residue Field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ring and \mathfrak is then its unique maximal ideal. In abstract algebra, the splitting field of a polynomial is constructed using residue fields. Residue fields also applied in algebraic geometry, where to every point x of a scheme X one associates its residue field k(x). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the ''natural domain'' for the coordinates of the point. Definition Suppose that R is a commutative local ring, with maximal ideal \mathfrak. Then the residue field is the quotient ring R/\mathfrak. Now suppose that X is a scheme and x is a point of X. By the definition of a scheme, we may find an affine neighbourhood \mathcal = \text(A) of x, with some commutative ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
