Simple extension

In mathematics, more specifically 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 extensions which are simple.

Definition

A field extension L/K is called a simple extension if there exists an element θ in L with

$L=K(\theta ).$

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.

A primitive element of a finite field is a generator of the field's multiplicative group. When said at greater length: In the realm of finite fields, a stricter definition of primitive element is used. The multiplicative group of a finite field is cyclic, and an element is called a primitive element if and only if it is a generator for the multiplicative group. The distinction is that the earlier definition requires that every element of the field be a quotient of polynomials in the primitive element, but within the realm of finite fields the requirement is that every nonzero element be a pure power of a primitive element. An attempt to distinguish these meanings by calling θ a field primitive element of L over K, and a generator of the multiplicative group of a finite field a group primitive element is made by (Roman 1995).

Notes

The only field contained in L which contains both K and θ is L itself. More concretely, 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).

K(θ) is defined as the smallest field which contains K[θ], the polynomials in θ. As K[θ] is an integral domain this is the field of fractions of K[θ] and thus

$K(\theta )=\left\{\left.{\frac {g}{h}}\right\vert g,h\in K[\theta ],h\neq 0\right\}.$

In other words every element of K(θ) can be written as a quotient of two polynomials in θ with coefficients from K.

Examples

• C:R (generated by i)
• Q(√2):Q (generated by √2), more generally any number field (i.e., a finite extension of Q) is a simple extension Q(α) for some α. For example, $\mathbf {Q} ({\sqrt {3}},{\sqrt {7}})$ is generated by ${\sqrt {3}}+{\sqrt {7}}$.
• F(X):F (generated by X).

Classification of simple extensions

Given a field K the simple extensions K(θ) can be completely classified using the polynomial ring K[X] in one indeterminate,

Let K(θ) be a simple extension. If θ is algebraic over K then K(θ) is identical to K[θ]. If θ is transcendental over K then K(θ) is isomorphic to the field of fractions of K[X].

References

• Roman, Steven (1995). Field Theory. Graduate Texts in Mathematics. Vol. 158. New York: Springer-Verlag. ISBN 0-387-94408-7. Zbl 0816.12001.