Simple extension

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. 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. A field extension L/K is called a simple extension if there exists an element θ in L with The element θ is called a primitive element, or generating element, for the extension; we also say that L is generated over K by θ. Every finite field is a simple extension of the prime field of the same characteristic. More precisely, if p is a prime number and q = p d {displaystyle q=p^{d}} the field F q {displaystyle mathbb {F} _{q}} of q elements is a simple extension of degree d of F p . {displaystyle mathbb {F} _{p}.} This means that it is generated by an element θ that is a root of an irreducible polynomial of degree d. However, in this case, θ is normally not referred to as a primitive element, even though it fits the definition given in the previous paragraph. The reason is that in the case of finite fields, there is a competing definition of primitive element. Indeed, a primitive element of a finite field is usually defined as a generator of the field's multiplicative group. More precisely, by little Fermat theorem, the nonzero elements of F q {displaystyle mathbb {F} _{q}} (i.e. its multiplicative group) are the roots of the equation that is the (q−1)-th roots of unity. Therefore, in this context, a primitive element is a primitive (q−1)-th root of unity, that is a generator of the multiplicative group of the nonzero elements of the field. Clearly, a group primitive element is a field primitive element, but the contrary is false. Thus the general definition requires that every element of the field may be expressed as a polynomial in the generator, while, in the realm of finite fields, every nonzero element of the field is a pure power of the primitive element. To distinguish these meanings one may use field primitive element of L over K for the general notion, and group primitive element for the finite field notion. If L is a simple extension of K generated by θ, it is the only field contained in L 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).