| Simple Extension |
Article Index for Simple |
Website Links For Simple |
Information AboutSimple Extension |
| CATEGORIES ABOUT SIMPLE EXTENSION | |
| field theory | |
|
The Primitive Element Theorem provides a characterization of the Finite extensions which are simple. DEFINITION A field extension ''L''/''K'' is called a simple extension if there exists an element θ in ''L'' with : The element θ is called a primitive element, or '''generating element''', for the extension; we also say that ''L'' is '''generated over''' ''K'' by θ. NOTES The only field contained in ''L'' which contains both ''K'' and θ is ''L'' itself. More concretely, this means that every element of ''L'' can be obtained from the elements of ''K'' and θ by finitely many field operations (addition, subtraction, multiplication and division). ''K''(θ) is defined as the smallest fields which contains ''K'' the polynomials in θ. As ''K''[θ is an Integral Domain this is the Field Of Fractions of ''K''[θ] and thus |
|
|