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Concept
Simple extension
Summary
In field theory, a simple extension is a field extension which 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.
A field extension L/K is called a simple extension if there exists an element θ in L with
This means that every element of L can be expressed as a rational fraction in θ, with coefficients in K; that is, it is produced from θ and elements of K by the field operations +, −, •, / . Equivalently, L is the smallest field which contains both K and θ.
There are two different kinds of simple extensions (see Structure of simple extensions below).
The element θ may be transcendental over K, which means that it is not a root of any polynomial with coefficients in K. In this case is isomorphic to the field of rational functions
Otherwise, θ is algebraic over K; that is, θ is a root of a polynomial over K. The monic polynomial of minimal degree n, with θ as a root, is called the minimal polynomial of θ. Its degree equals the degree of the field extension, that is, the dimension of L viewed as a K-vector space. In this case, every element of can be uniquely expressed as a polynomial in θ of degree less than n, and is isomorphic to the quotient ring
In both cases, the element θ is called a generating element or primitive element for the extension; one says also L is generated over K by θ.
For example, every finite field is a simple extension of the prime field of the same characteristic. More precisely, if p is a prime number and the field of q elements is a simple extension of degree n of In fact, L is generated as a field by any element θ that is a root of an irreducible polynomial of degree n in .
However, in the case of finite fields, the term primitive element is usually reserved for a stronger notion, an element γ which generates as a multiplicative group, so that every nonzero element of L is a power of γ, i.e.
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