Simple extension explained
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 is called a simple extension if there exists an element in L with
This means that every element of can be expressed as a rational fraction in, with coefficients in ; that is, it is produced from and elements of by the field operations +, −, •, / . Equivalently, is the smallest field that contains both
and .
There are two different kinds of simple extensions (see Structure of simple extensions below).
The element may be transcendental over, which means that it is not a root of any polynomial with coefficients in . In this case
is isomorphic to the field of rational functions
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
can be uniquely expressed as a polynomial in of degree less than, and
is isomorphic to the
quotient ring
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
the field
of elements is a simple extension of degree
n of
In fact,
L is generated as a field by any element that is a root of an
irreducible polynomial of
degree n in
.
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 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{align}
\varphi:K[X]& → L\\
f(X)&\mapstof(\theta).
\end{align}
Two cases may occur.
If
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
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
. That is, we have a
K-basis of
L given by
1,\theta,\theta2,\ldots,\thetan-1
.
Examples
.
) /
Q generated by
.
- Any number field (i.e., a finite extension of Q) is a simple extension Q(θ) for some θ. For example,
is generated by
.
- 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
- Book: Roman, Steven . Steven Roman
. Steven Roman . Field Theory . . 158 . . New York . 1995 . 0-387-94408-7 . 0816.12001 .
