Simple extension

In field theory, a simple extension is a field extension that is generated by the adjunction of a single element, called a primitive 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 Template:Math is called a simple extension if there exists an element Template:Math in L with

$L=K(\theta ).$

This means that every element of Template:Mvar can be expressed as a rational fraction in Template:Math, with coefficients in Template:Mvar; that is, it is produced from Template:Math and elements of Template:Mvar by the field operations +, −, •, / . Equivalently, Template:Mvar is the smallest field that contains both Template:Mvar and Template:Math.

There are two different kinds of simple extensions (see Structure of simple extensions below).

The element Template:Math may be transcendental over Template:Mvar, which means that it is not a root of any polynomial with coefficients in Template:Mvar. In this case $K(\theta )$ is isomorphic to the field of rational functions $K(X).$

Otherwise, Template:Math is algebraic over Template:Mvar; that is, Template:Math is a root of a polynomial over Template:Mvar. The monic polynomial $p(X)$ of minimal degree Template:Mvar, with Template:Math as a root, is called the minimal polynomial of Template:Math. Its degree equals the degree of the field extension, that is, the dimension of Template:Mvar viewed as a Template:Mvar-vector space. In this case, every element of $K(\theta )$ can be uniquely expressed as a polynomial in Template:Math of degree less than Template:Mvar, and $K(\theta )$ is isomorphic to the quotient ring $K[X]/(p(X)).$

In both cases, the element Template:Math is called a generating element or primitive element for the extension; one says also Template:Math is generated over Template:Math by Template:Math.

For example, every finite field is a simple extension of the prime field of the same characteristic. More precisely, if Template:Math is a prime number and $q=p^n,$ the field $L={𝔽}_q$ of Template:Math elements is a simple extension of degree n of $K={𝔽}_p.$ In fact, L is generated as a field by any element Template:Math that is a root of an irreducible polynomial of degree n in $K[X]$.

However, in the case of finite fields, the term primitive element is usually reserved for a stronger notion, an element γ that generates $L^{\times }=L-\{0\}$ as a multiplicative group, so that every nonzero element of L is a power of γ, i.e. is produced from γ using only the group operation • . To distinguish these meanings, one uses the term "generator" or field primitive element for the weaker meaning, reserving "primitive element" or group primitive element for the stronger meaning. (See Template:Slink and Primitive element (finite field)).

Structure of simple extensions

Let L be a simple extension of K generated by θ. For the polynomial ring K[X], one of its main properties is the unique ring homomorphism

$\begin{array}{cc}\phi :K[X]& \to L\\ f(X)& \mapsto f(\theta ).\end{array}$

Two cases may occur.

If $\phi$ is injective, it may be extended injectively to the field of fractions K(X) of K[X]. Since L is generated by θ, this implies that $\phi$ 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 $\phi$ is not injective, let p(X) be a generator of its kernel, which is thus the minimal polynomial of θ. The image of $\phi$ is a subring of L, and thus an integral domain. This implies that p is an irreducible polynomial, and thus that the quotient ring $K[X]/⟨p(X)⟩$ is a field. As L is generated by θ, $\phi$ is surjective, and $\phi$ induces an isomorphism from $K[X]/⟨p(X)⟩$ onto L. This implies that every element of L is equal to a unique polynomial in θ of degree lower than the degree $n=\mathrm{deg}p(X)$. That is, we have a K-basis of L given by $1,\theta ,{\theta }^2,\dots ,{\theta }^{n-1}$.

Examples

• C / R generated by $\theta =i=\sqrt{-1}$.
• Q($\sqrt{2}$) / Q generated by $\theta =\sqrt{2}$.
• Any number field (i.e., a finite extension of Q) is a simple extension Q(θ) for some θ. For example, $𝐐(\sqrt{3},\sqrt{7})$ is generated by $\theta =\sqrt{3}+\sqrt{7}$.
• F(X) / F, a field of rational functions, is generated by the formal variable X.

See also

• Companion matrix for the multiplication map on a simple field extension

Literature

• Template:Cite book