# 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

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 the field of *q* elements is a simple extension of degree *d* of This means that it is generated by an element *θ* that is a root of an irreducible polynomial of degree *d*. However, in this case, *θ* is normally not referred to as a *primitive element*, even though it fits the definition given in the previous paragraph.

The reason is that in the case of finite fields, there is a competing definition of primitive element. Indeed, 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 (i.e. its multiplicative group) are the roots of the equation

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 converse 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* then it is the smallest field 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).

Consider the polynomial ring *K*[*X*]. One of its main properties is that there exists a unique ring homomorphism

Two cases may occur.

If 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 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 is not injective, let *p*(X) be a generator of its kernel, which is thus the minimal polynomial of *θ*. The image of is a subring of *L*, and thus an integral domain. This implies that *p* is an irreducible polynomial, and thus that the quotient ring is a field. As *L* is generated by *θ*, is surjective, and induces an isomorphism from 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**():**Q**(generated by ), more generally any number field (i.e., a finite extension of**Q**) is a simple extension**Q**(*α*) for some*α*. For example, is generated by .*F*(*X*):*F*(generated by*X*).

## References

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