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.

Every finite field is a simple extension of the prime field of the same characteristic. More precisely, if p is a prime number and $q=p^{d}$ the field $\mathbb {F} _{q}$ of q elements is a simple extension of degree d of $\mathbb {F} _{p}.$ This means that it is generated by an element θ which is a root of an irreducible polynomial of degree d. However, in this case, θ is normally not referred to as a primitive element.

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 elements of $\mathbb {F} _{q}$ are the roots of the equation

x^{q}-1=0,

that is the q-th roots of unity. Therefore, in this context, a primitive element is a primitive q-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.^[1]

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] and is isomorphic to the quotient ring of the polynomial ring K[X] by (the ideal generated) by the minimal polynomial of θ. If θ is transcendental over K then K(θ) is isomorphic to the field of fractions of K(X) 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.

^ (Roman 1995)

[1] (Roman 1995)

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