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Regular 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] |
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] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraically Closed
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] |
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] |
Linearly Disjoint
In mathematics, algebras ''A'', ''B'' over a field ''k'' inside some field extension \Omega of ''k'' are said to be linearly disjoint over ''k'' if the following equivalent conditions are met: *(i) The map A \otimes_k B \to AB induced by (x, y) \mapsto xy is injective. *(ii) Any ''k''-basis of ''A'' remains linearly independent over ''B''. *(iii) If u_i, v_j are ''k''-bases for ''A'', ''B'', then the products u_i v_j are linearly independent over ''k''. Note that, since every subalgebra of \Omega is a domain, (i) implies A \otimes_k B is a domain (in particular reduced). Conversely if ''A'' and ''B'' are fields and either ''A'' or ''B'' is an algebraic extension of ''k'' and A \otimes_k B is a domain then it is a field and ''A'' and ''B'' are linearly disjoint. However, there are examples where A \otimes_k B is a domain but ''A'' and ''B'' are not linearly disjoint: for example, ''A'' = ''B'' = ''k''(''t''), the field of rational functions over ''k''. One also has: ''A'', ''B'' are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Extension
In field theory, a branch of algebra, a primary extension ''L'' of ''K'' is a field extension such that the algebraic closure of ''K'' in ''L'' is purely inseparable over ''K''.Fried & Jarden (2008) p.44 Properties * An extension ''L''/''K'' is primary if and only if it is linearly disjoint from the separable closure of ''K'' over ''K''. * A subextension of a primary extension is primary. * A primary extension of a primary extension is primary (transitivity). * Any extension of a separably closed field is primary. * An extension is regular if and only if it is separable and primary. * A primary extension of a 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 ... is regular. References * Field (mathematics) {{Abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Purely Transcendental Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Algebraic Geometry
''Foundations of Algebraic Geometry'' is a book by that develops algebraic geometry over field (mathematics), fields of any characteristic (algebra), characteristic. In particular it gives a careful treatment of intersection theory by defining the local intersection multiplicity of two Subvariety, subvarieties. Weil was motivated by the need for a rigorous theory of correspondences on algebraic curves in positive characteristic, which he used in his proof of the Riemann hypothesis for curves over a finite field. Weil introduced Abstract variety, abstract rather than Projective variety, projective varieties partly so that he could construct the Jacobian of a curve. (It was not known at the time that Jacobians are always projective varieties.) It was some time before anyone found any examples of complete variety, complete abstract varieties that are not projective. In the 1950s Weil's work was one of several competing attempts to provide satisfactory foundations for algebraic g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |