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 is called a simple extension if there exists an element in ''L'' with
:$L = K(\theta).$
This means that every element of can be expressed as a rational fraction in , with coefficients in .

There are two different sort of simple extensions.

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

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

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

For example, every finite field is a simple extension of the prime field of the same characteristic. More precisely, if is a prime number and $q=p^d,$ the field $\mathbb_$ of elements is a simple extension of degree of $\mathbb_.$ This means that it is generated by an element that is a root of an irreducible polynomial of degree .

However, in the case of finite fields, is normally not referred to as a ''primitive element'', even though it fits the definition given above.
The reason is that in the case of finite fields, there is a competing definition of a primitive element. Indeed, a primitive element of a finite field is an element such that every nonzero element of the field is a power of ; that is, a generator of the field's multiplicative group (see and Primitive element (finite field) for details). A primitive element (in this sense) is a generating element (since an integer power is a special case of a polynomial), but the converse is false.

In summary, the general definition of a simple extension requires that all elements of the field can be expressed as a polynomials in a single ''generator'', which is also called a ''primitive element''. In the case of finite fields, every extension is simple, and all nonzero elements are pure powers of single element that is also called a ''primitive element''. To distinguish these meanings in the realm of finite fields, one uses generally the term "generator" for the first meaning, and one reserves "primitive element" for the second meaning. The terms of field primitive element for the first notion, and group primitive element for the second one are sometimes used.

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
:
\begin
\varphi: K &\rightarrow L\\
p(X) &\mapsto p(\theta)\,.
\end

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 \langle p \rangle is a field. As ''L'' is generated by ''θ'', $\varphi$ is surjective, and $\varphi$ induces an isomorphism from K \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($\sqrt$):Q (generated by $\sqrt$), more generally any number field (i.e., a finite extension of Q) is a simple extension Q(''α'') for some ''α''. For example, $\mathbf(\sqrt, \sqrt)$ is generated by $\sqrt + \sqrt$.
* ''F''(''X''):''F'' (generated by ''X'').

References

*{{cite book , last = Roman , first = Steven , authorlink=Steven Roman , title = Field Theory , series=Graduate Texts in Mathematics , volume=158 , publisher = Springer-Verlag , place = New York , year = 1995 , isbn = 0-387-94408-7 , zbl=0816.12001

Field extensions