of the field can be expressed as a polynomials in a single generator, which is also called a primitive element. In the case of finite fields, every extension is simple, and all nonzero elements are pure powers of single element that is also called a primitive element. To distinguish these meanings in the realm of finite fields, one uses generally the term "generator" for the first meaning, and one reserves "primitive element" for the second meaning. The terms of field primitive element for the first notion, and group primitive element for the second one are sometimes used. Structure of simple extensions If L is a simple extension of K generated by θ then it is the smallest field which contains both K and θ. 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). Consider the polynomial ring K[X]. One of its main properties is that there exists a unique ring homomorphism Two cases may occur. If is injective, it may be extended to the field of fractions K(X) of K[X]. As we have supposed that 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 of the extension. Examples C:R (generated by i) Q():Q (generated by ), more generally any number field (i.e., a finite extension of Q) is a simple extension Q(α) for some α. For example, is generated by . F(X):F (generated by X). References Field extensions