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:$L\; =\; K(\; heta).$

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.

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

:$K(\; heta)\; =\; left\; \{\; frac\{g\}\{h\}\; |\; g,h\; in\; K\; [\; heta]\; ,\; h\; eq\; 0\; ight\; \}.$

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 is a simple extension Q(α) for some α
* "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

See also

* Primitive element theorem