Simple extension

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.

Definition 
A field extension  is called a simple extension if there exists an element  in L with

This means that every element of  can be expressed as a rational fraction in , with coefficients in ; that is, it is produced from  and elements of  by the field operations +, −, •, / . Equivalently,  is the smallest field which contains both  and .

There are two different kinds of simple extensions (see Structure of simple extensions below).

The element  may be transcendental over , which means that it is not a root of any polynomial with coefficients in . In this case  is isomorphic to the field of rational functions 

Otherwise,  is algebraic over ; that is,  is a root of a polynomial over . The monic polynomial  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 dimension of  viewed as a -vector space. In this case, every element of  can be uniquely expressed as a polynomial in  of degree less than , and  is isomorphic to the quotient ring 

In both cases, the element  is called a generating element or primitive element for the extension; one says also  is generated over  by .

For example, every finite field is a simple extension of the prime field of the same characteristic. More precisely, if  is a prime number and  the field  of  elements is a simple extension of degree n of  In fact, L is generated as a field by any element  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. 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. (See  and Primitive element (finite field)).

Structure of simple extensions 
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.

If  is injective, it may be extended to the field of fractions K(X) of K[X]. Since L is generated by θ, this implies that  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. The image of  is a subring of L, and thus an integral domain. This implies that p is an irreducible polynomial, and thus that the quotient ring  is a field. As L is generated by θ,  is surjective, and  induces an isomorphism from  onto L. This implies that every element of L is equal to a unique polynomial in θ of degree lower than the degree . That is, we have a K-basis of L given by .

Examples 
 C:R generated by .
 Q():Q generated by .
 Any number field (i.e., a finite extension of Q) is a simple extension Q(θ) for some θ. For example,  is generated by .
 F(X):F, a field of rational functions, is generated by the formal variable X.

See also 

 Companion matrix for the multiplication map on a simple field extension

References

Literature

Field extensions