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Field Homomorphism
Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative ring in which and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division. The non-zero elements of a field ''F'' form an abelian group under multiplication; this group is typically denoted by ''F''×; The ring of polynomials in the variable ''x'' with coefficients in ''F'' is denoted by ''F'' 'x'' Basic definitions ; Characteristic : The ''characteristic'' of the field ''F'' is the smallest positive integer ''n'' such that ; here ''n''·1 stands for ''n'' summands . If no such ''n'' exists, we say the characteristic is zero. Every non-zero characteristic is a prime number. For example, the rational numbers, the real numbers and the ''p' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Theory (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] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field ''K'' is unique up to an isomorphism that fixes every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''. The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''. To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''. The algebraic closure of ''K'' is also the smallest algebraically closed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed Field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it. Every field K is contained in an algebraically closed field C, and the roots in C of the polynomials with coefficients in K form an algebraically closed field called an algebraic closure of K. Given two algebraic closures of K there are isomorphisms between them that fix the elements of K. Algebraically closed fields appear in the following chain of class inclusions: Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation x^2+1=0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imperfect Degree
The imperfect (abbreviated ) is a verb form that combines past tense (reference to a past time) and imperfective aspect (reference to a continuing or repeated event or state). It can have meanings similar to the English "was doing (something)" or "used to do (something)". It contrasts with preterite forms, which refer to a single completed event in the past. Traditionally, the imperfect of languages such as Latin and French is referred to as one of the tenses, although it actually encodes aspectual information in addition to tense (time reference). It may be more precisely called ''past imperfective''. English has no general imperfective and expresses it in different ways. The term "imperfect" in English refers to forms much more commonly called '' past progressive'' or ''past continuous'' (e.g. "was doing" or "were doing"). These are combinations of past tense with specifically continuous or progressive aspect. In German, formerly referred to the simply conjugated past tense ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Field
In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has no multiple roots in any field extension ''F/k''. * Every irreducible polynomial over ''k'' has non-zero formal derivative. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' is separable. * Every algebraic extension of ''k'' is separable. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic , every element of ''k'' is a ''p''th power. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic , the Frobenius endomorphism is an automorphism of ''k''. * The separable closure of ''k'' is algebraically closed. * Every reduced commutative ''k''-algebra ''A'' is a separable algebra; i.e., A \otimes_k F is reduced for every field extension ''F''/''k''. (see below) Otherwise, ''k'' is called imperfect. In particular, all fields of characteristic zero and all finite fields ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Polynomial
In mathematics, a polynomial ''P''(''X'') over a given field ''K'' is separable if its roots are distinct in an algebraic closure of ''K'', that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely related to square-free polynomial. If ''K'' is a perfect field then the two concepts coincide. In general, ''P''(''X'') is separable if and only if it is square-free over any field that contains ''K'', which holds if and only if ''P''(''X'') is coprime to its formal derivative ''D'' ''P''(''X''). Older definition In an older definition, ''P''(''X'') was considered separable if each of its irreducible factors in ''K'' 'X''is separable in the modern definition. In this definition, separability depended on the field ''K''; for example, any polynomial over a perfect field would have been considered separable. This definition, although it can be convenient for Galois theory, is no longer in use. Separable field extensions Separable pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Extension
In field theory (mathematics), field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial (field theory), minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated zero of a function, roots in any extension field).Isaacs, p. 281 There is also a more general definition that applies when is not necessarily algebraic over . An extension that is not separable is said to be ''inseparable''. Every algebraic extension of a field (mathematics), field of characteristic (algebra)#Case of fields, characteristic zero is separable, and every algebraic extension of a finite field is separable.Isaacs, Theorem 18.11, p. 281 It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Extension
In abstract algebra, a normal extension is an Algebraic extension, algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' that has a zero of a function, root in ''L'' splits into linear factors over ''L''. This is one of the conditions for an algebraic extension to be a Galois extension. Nicolas Bourbaki, Bourbaki calls such an extension a quasi-Galois extension. For Finite extension, finite extensions, a normal extension is identical to a splitting field. Definition Let ''L/K'' be an algebraic extension (i.e., ''L'' is an algebraic extension of ''K''), such that L\subseteq \overline (i.e., ''L'' is contained in an algebraic closure of ''K''). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent: * Every Embedding (field theory), embedding of ''L'' in \overline over ''K'' induces an automorphism of ''L''. * ''L'' is the splitting field of a family of polynomials in K[X]. * Every irreducibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting Field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a polynomial ''p''(''X'') over a field ''K'' is a field extension ''L'' of ''K'' over which ''p'' factors into linear factors :p(X) = c \prod_^ (X - a_i) where c \in K and for each i we have X - a_i \in L /math> with ''ai'' not necessarily distinct and such that the roots ''ai'' generate ''L'' over ''K''. The extension ''L'' is then an extension of minimal degree over ''K'' in which ''p'' splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom in that isomorphism is known as the Galois group of ''p'' (if we assume it is separable). A splitting field of a set ''P'' of polynomials is the smallest field over which each of the polynomials in ''P'' splits. Properties An extension ''L'' th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Extension
In field theory, a simple extension is a field extension that is generated by the adjunction of a single element, called a ''primitive element''. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of the finite simple extensions. Definition A field extension is called a simple extension if there exists an element in ''L'' with :L = K(\theta). This means that every element of can be expressed as a rational fraction in , with coefficients in ; that is, it is produced from and elements of by the field operations +, −, •, / . Equivalently, is the smallest field that contains both ' and . There are two different kinds of simple extensions (see below): # The element may be transcendental over , which means that it is not a root of any polynomial with coefficients in . In this case K(\theta) is isomorphic to the field of rational functions K(X). # Otherwise, is algebraic over ; that is, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Element (field Theory)
In field theory (mathematics), field theory, a simple extension is a field extension that is generated by the adjunction (field theory), adjunction of a single element, called a ''primitive element''. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of the finite extension, finite simple extensions. Definition A field extension is called a simple extension if there exists an element in ''L'' with :L = K(\theta). This means that every element of can be expressed as a rational fraction in , with coefficients in ; that is, it is produced from and elements of by the field operations +, −, •, / . Equivalently, is the smallest field that contains both ' and . There are two different kinds of simple extensions (see below): # The element may be transcendental element, transcendental over , which means that it is not a zero of a function, root of any polynomial with coefficients in . In this case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
