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.
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The primitive element theorem provides a characterization of the finite simple extensions.
Definition
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
In fact, 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
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.
Structure of simple extensions
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).
Let us consider the polynomial ring K[X]. One of its main properties is that there exists a unique ring homomorphism
Two cases may occur.
If
If