Algebraic extension

Some properties

All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic. The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.

Let E be an extension field of K, and a ∈ E. The smallest subfield of E that contains K and a is commonly denoted K(a). If a is algebraic over K, then the elements of K(a) can be expressed as polynomials in a with coefficients in K; that is, K(a)=K[a], the smallest ring containing K and a. In this case, K(a) is a finite extension of K and all its elements are algebraic over K. In particular, K(a) is a K-vector space with basis {1,a,...,a^{d-1}}, where d is the degree of the minimal polynomial of a. These properties do not hold if a is not algebraic. For example, \Q(\pi)\neq \Q[\pi], and they are both infinite dimensional vector spaces over \Q.

An algebraically closed field F has no proper algebraic extensions, that is, no algebraic extensions E with F

An extension L/K is algebraic if and only if every sub K-algebra of L is a field.

Properties

The following three properties hold:

1. If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K.
2. If E and F are algebraic extensions of K in a common overfield C, then the compositum EF is an algebraic extension of K.
3. 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:

| The union of any chain of algebraic extensions over a base field is itself an algebraic extension over the same base field.

This fact, together with Zorn's lemma (applied to an appropriately chosen poset), establishes the existence of algebraic closures.

Generalizations

Main article: Substructure (mathematics)

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.

Relative algebraic closures

Given a field k and a field K containing k, one defines the relative algebraic closure of k in K to be the subfield of K consisting of all elements of K that are algebraic over k, that is all elements of K that are a root of some nonzero polynomial with coefficients in k.

Notes

References

References

1. Fraleigh (2014), Definition 31.1, p. 283.
2. Malik, Mordeson, Sen (1997), Definition 21.1.23, p. 453.
3. Fraleigh (2014), Definition 29.6, p. 267.
4. Malik, Mordeson, Sen (1997), Theorem 21.1.8, p. 447.
5. See also Hazewinkel et al. (2004), p. 3.
6. Fraleigh (2014), Theorem 31.18, p. 288.
7. Fraleigh (2014), Corollary 31.13, p. 287.
8. Fraleigh (2014), Theorem 30.23, p. 280.
9. Fraleigh (2014), Example 29.8, p. 268.
10. Fraleigh (2014), Corollary 31.16, p. 287.
11. Fraleigh (2014), Theorem 31.22, p. 290.
12. Lang (2002) p.228