Algebraic extension

In mathematics, an algebraic extension is a field extension Template:Math such that every element of the larger field Template:Mvar is algebraic over the smaller field Template:Mvar; that is, every element of Template:Mvar is a root of a non-zero polynomial with coefficients in Template:Mvar.<ref>Fraleigh (2014), Definition 31.1, p. 283.</ref><ref>Malik, Mordeson, Sen (1997), Definition 21.1.23, p. 453.</ref> A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic.<ref>Fraleigh (2014), Definition 29.6, p. 267.</ref><ref>Malik, Mordeson, Sen (1997), Theorem 21.1.8, p. 447.</ref>

The algebraic extensions of the field <math>\Q</math> of the rational numbers are called algebraic number fields and are the main objects of study of algebraic number theory. Another example of a common algebraic extension is the extension <math>\Complex/\R</math> of the real numbers by the complex numbers.

Some properties

All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic.<ref>See also Hazewinkel et al. (2004), p. 3.</ref> The converse is not true however: there are infinite extensions which are algebraic.<ref>Fraleigh (2014), Theorem 31.18, p. 288.</ref> For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.<ref>Fraleigh (2014), Corollary 31.13, p. 287.</ref>

Let Template:Math be an extension field of Template:Math, and Template:Math. The smallest subfield of Template:Math that contains Template:Math and Template:Mvar is commonly denoted <math>K(a).</math> If Template:Mvar is algebraic over Template:Math, then the elements of Template:Math can be expressed as polynomials in Template:Mvar with coefficients in K; that is, <math>K(a)=K[a]</math>, the smallest ring containing Template:Math and Template:Mvar. In this case, <math>K(a)</math> is a finite extension of Template:Mvar and all its elements are algebraic over Template:Mvar. In particular, <math>K(a)</math> is a Template:Mvar-vector space with basis <math>\{1,a,...,a^{d-1}\}</math>, where d is the degree of the minimal polynomial of Template:Mvar.<ref>Fraleigh (2014), Theorem 30.23, p. 280.</ref> These properties do not hold if Template:Mvar is not algebraic. For example, <math>\Q(\pi)\neq \Q[\pi],</math> and they are both infinite dimensional vector spaces over <math>\Q.</math><ref>Fraleigh (2014), Example 29.8, p. 268.</ref>

An algebraically closed field F has no proper algebraic extensions, that is, no algebraic extensions E with F < E.<ref>Fraleigh (2014), Corollary 31.16, p. 287.</ref> 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.<ref>Fraleigh (2014), Theorem 31.22, p. 290.</ref>

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:<ref>Lang (2002) p.228</ref>

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:

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

Generalizations

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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

<math>\left\{y\in N \mid p(y)\right\}</math>

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.

See also

• Integral element
• Lüroth's theorem
• Galois extension
• Separable extension
• Normal extension

Notes

<references/>

References

• Template:Citation
• Template:Citation
• Template:Lang Algebra
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation