In field theory, a simple extension is a field extension that 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 that contains both K and θ.
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
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.
can be uniquely expressed as a polynomial in θ of degree less than n, and
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 that generates
as a multiplicative group, so that every nonzero element of L is a power of γ, i.e. is produced from γ using only the group operation • .
To distinguish these meanings, one uses the term "generator" or field primitive element for the weaker meaning, reserving "primitive element" or group primitive element for the stronger meaning.
[1] (See Finite field § Multiplicative structure and Primitive element (finite field)).
Let L be a simple extension of K generated by θ.
For the polynomial ring K[X], one of its main properties is the unique ring homomorphism Two cases may occur.
is an isomorphism from K(X) onto L. This implies that every element of L is equal to an irreducible fraction of polynomials in θ, and that two such irreducible fractions are equal if and only if one may pass from one to the other by multiplying the numerator and the denominator by the same non zero element of K. If
is not injective, let p(X) be a generator of its kernel, which is thus the minimal polynomial of θ.
This implies that p is an irreducible polynomial, and thus that the quotient ring
onto L. This implies that every element of L is equal to a unique polynomial in θ of degree lower than the degree