In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K.Fraleigh (2014), Definition 31.1, p. 283.Malik, Mordeson, Sen (1997), Definition 21.1.23, p. 453.
Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental.Fraleigh (2014), Definition 29.6, p. 267.Malik, Mordeson, Sen (1997), Theorem 21.1.8, p. 447.
For example, the field extension R/Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental,Malik, Mordeson, Sen (1997), Example 21.1.17, p. 451.
while the field extensions C/RMalik, Mordeson, Sen (1997), Theorem 21.1.8, p. 447.
and Q()/QFraleigh (2014), Example 31.8, p. 285.
are algebraic, where C is the field of complex numbers.
All transcendental extensions are of infinite degree.
This in turn implies that all finite extensions are algebraic.See also Hazewinkel et al. (2004), p.
3. The converse is not true however: there are infinite extensions which are algebraic.Fraleigh (2014), Theorem 31.18, p. 288.
For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.Fraleigh (2014), Corollary 31.13, p. 287.
Let E be an extension field of K, and a ∈ E.
If a is algebraic over K, then K(a), the set of all polynomials in a with coefficients in K, is not only a ring but a field: K(a) is an algebraic extension of K which has finite degree over K.Fraleigh (2014), Theorem 30.23, p. 280.
The converse is not true.
Q[π] and Q[e] are fields but π and e are transcendental over Q.Fraleigh (2014), Example 29.8, p. 268.
An algebraically closed field F has no proper algebraic extensions, that is, no algebraic extensions E with F < E.Fraleigh (2014), Corollary 31.16, p. 287.
An example is the field of complex numbers.
Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice.Fraleigh (2014), Theorem 31.22, p. 290.
An extension L/K is algebraic if and only if every sub K-algebra of L is a field.
Properties
The class of algebraic extensions forms a distinguished class of field extensions, that is, the following three properties hold:Lang (2002) p.228
If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K.
If E and F are algebraic extensions of K in a common overfield C, then the compositum EF is an algebraic extension of K.
If E is an algebraic extension of F and E>K>F then E is an algebraic extension of K.
These finitary results can be generalized using transfinite induction:
This fact, together with Zorn's lemma (applied to an appropriately chosen poset), establishes the existence of algebraic closures.
Generalizations
Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p(x) is true and the set
\left\{y\in N \mid p(y)\right\}
is finite.
It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension.
The Galois group of N over M can again be defined as the group of automorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case.
See also
Integral element
Lüroth's theorem
Galois extension
Separable extension
Normal extension
Notes
References
