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Glossary Of Field Theory
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 (''F'',+,*) in which 0≠1 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 ''n''·1 = 0; here ''n''·1 stands for ''n'' summands 1 + 1 + 1 + ... + 1. If no such ''n'' exists, we say the characteristic is zero. Every non-zero characteristic is a prime number. For example, the rationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Theory (mathematics)
Field theory may refer to: Science * Field (mathematics), the theory of the algebraic concept of field * Field theory (physics), a physical theory which employs fields in the physical sense, consisting of three types: ** Classical field theory, the theory and dynamics of classical fields ** Quantum field theory, the theory of quantum mechanical fields ** Statistical field theory, the theory of critical phase transitions **Grand unified theory Social science * Field theory (psychology) Field theory is a psychological theory (more precisely: Topological and vector psychology) which examines patterns of interaction between the individual and the total field, or environment. The concept first made its appearance in psychology with r ..., a psychological theory which examines patterns of interaction between the individual and his or her environment * Field theory (sociology), a sociological theory concerning the relationship between social actors and local social orders {{Disambig ... [...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 whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...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 fiel ... [...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 . 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 closed. Also, no finite field ''F'' is algebraically closed, because if ''a''1, ''a''2, ..., ''an'' are the elements of ''F'', then the polynomial (''x'' − ''a''1)(''x'' − ''a''2) ⋯ (''x'' − ''a''''n'') + 1 has no zero in ''F''. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraicall ... [...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 walking" or "used to walk". 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 (to contrast with the ... [...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 distinct roots. * 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 are perfect. Perfect fields are significant because Galois theory over these fields becomes simpler, since the ... [...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.N. Jacobson, Basic Algebra I, p. 233 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. Separabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Extension
In 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 of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated 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 of 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 obstacle for extending many theorems proved in characteristic zero to non-zero characteristic. For example, the fundamental theorem of Galois ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Extension
In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic extensions to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension. 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 of ''L'' in \overline induces an automorphism of ''L''. * ''L'' is the splitting field of a family of polynomials in K\left \right/math>. * Every irreducible polynomial of K\left \right/math> which has a root in ''L'' splits into linear factors in ''L''. Other properties Let ''L'' be an extension of a field ''K''. Then: * ... [...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). Properties An extension ''L'' which is a splitting field for a set of polynomials ''p''(''X'') over ''K'' is called a normal extension of ''K''. Given an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Extension
In field theory, a simple extension is a field extension which is generated by the adjunction of a single 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 . There are two different sort of simple extensions. 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, is a root of a polynomial over . The monic polynomial F(X) of minimal degree , with as a root, is called the minimal polynomial of . Its degree equals the degree of the field extension, that is, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Element (field Theory)
In field theory, a simple extension is a field extension which is generated by the adjunction of a single 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 . There are two different sort of simple extensions. 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, is a root of a polynomial over . The monic polynomial F(X) of minimal degree , with as a root, is called the minimal polynomial of . Its degree equals the degree of the field extension, that is, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
