In field theory, a simple extension is a field extension which is generated by the adjunction of a single 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 . There are two different sort of simple extensions. 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 of This means that it is generated by an element that is a root of an irreducible polynomial of degree . However, in the case of finite fields, is normally not referred to as a primitive element, even though it fits the definition given above. The reason is that in the case of finite fields, there is a competing definition of a primitive element. Indeed, a primitive element of a finite field is an element such that every nonzero element of the field is a power of ; that is, a generator of the field's multiplicative group (see and Primitive element (finite field) for details). A primitive element (in this sense) is a generating element (since an integer power is a special case of a polynomial), but the converse is false. In summary, the general definition of a simple extension requires that all elements