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.

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

$L = K(\theta).$

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 nonzero elements of $\mathbb{F}_{q}$ (i.e. its multiplicative group) are the roots of the equation

$x^{q-1}-1=0,$

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.^[1]

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

$\begin{align} \varphi: K[X] &\rightarrow L\\ p(X) &\mapsto p(\theta)\,. \end{align}$

Two cases may occur.

If $\varphi$ is injective, it may be extended to the field of fractions K(X) of K[X]. As we have supposed that L is generated by θ, this implies that $\varphi$ is an isomorphism from K(X) onto L. This implies that every element of L is equal to an irreducible fraction of polynomials in θ, and that two such irreducible fractions are equal if and only if one may pass from one to the other by multiplying the numerator and the denominator by the same non zero element of K.

If $\varphi$ is not injective, let p(X) be a generator of its kernel, which is thus the minimal polynomial of θ. The image of $\varphi$ is a subring of L, and thus an integral domain. This implies that p is an irreducible polynomial, and thus that the quotient ring $K[X]/\langle p \rangle$ is a field. As L is generated by θ, $\varphi$ is surjective, and $\varphi$ induces an isomorphism from $K[X]/\langle p \rangle$ onto L. This implies that every element of L is equal to a unique polynomial in θ, of degree lower than the degree of the extension.

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).

References

1. (Roman 1995)

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