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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] |
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] |