Algebraic extension

In mathematics, an algebraic extension is a field extension $L / K$ such that every element of the larger field $L$ is algebraic over the smaller field $K$ ; that is, every element of $L$ is a root of a non-zero polynomial with coefficients in $K$ .^[1]^[2] A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic.^[3]^[4]

The algebraic extensions of the field $\mathbb {Q}$ 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 $\mathbb {C} /\mathbb {R}$ 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.^[5] The converse is not true however: there are infinite extensions which are algebraic.^[6] For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.^[7]

Let $E$ be an extension field of $K$ , and $a \in E$ . 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, $K (a)$ is also the smallest ring containing $K$ and $a$ . In this case, $K(a)$ is a finite extension of $K$ (it is a finite dimensional $K$ -vector space), and all its elements are algebraic over $K$ .^[8] These properties do not hold if $a$ is not algebraic. For example, $\mathbb {Q} (\pi )\neq \mathbb {Q} [\pi ],$ and they are both infinite dimensional vector spaces over $\mathbb {Q} .$ ^[9]

An algebraically closed field F has no proper algebraic extensions, that is, no algebraic extensions E with F < E.^[10] 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.^[11]

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:^[12]

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:

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

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

^ Fraleigh (2014), Definition 31.1, p. 283.
^ Malik, Mordeson, Sen (1997), Definition 21.1.23, p. 453.
^ Fraleigh (2014), Definition 29.6, p. 267.
^ Malik, Mordeson, Sen (1997), Theorem 21.1.8, p. 447.
^ See also Hazewinkel et al. (2004), p. 3.
^ Fraleigh (2014), Theorem 31.18, p. 288.
^ Fraleigh (2014), Corollary 31.13, p. 287.
^ Fraleigh (2014), Theorem 30.23, p. 280.
^ Fraleigh (2014), Example 29.8, p. 268.
^ Fraleigh (2014), Corollary 31.16, p. 287.
^ Fraleigh (2014), Theorem 31.22, p. 290.
^ Lang (2002) p.228

References

Fraleigh, John B. (2014), A First Course in Abstract Algebra, Pearson, ISBN 978-1-292-02496-7
Hazewinkel, Michiel; Gubareni, Nadiya; Gubareni, Nadezhda Mikhaĭlovna; Kirichenko, Vladimir V. (2004), Algebras, rings and modules, vol. 1, Springer, ISBN 1-4020-2690-0
Lang, Serge (1993), "V.1:Algebraic Extensions", Algebra (Third ed.), Reading, Mass.: Addison-Wesley, pp. 223ff, ISBN 978-0-201-55540-0, Zbl 0848.13001
Malik, D. B.; Mordeson, John N.; Sen, M. K. (1997), Fundamentals of Abstract Algebra, McGraw-Hill, ISBN 0-07-040035-0
McCarthy, Paul J. (1991) [corrected reprint of 2nd edition, 1976], Algebraic extensions of fields, New York: Dover Publications, ISBN 0-486-66651-4, Zbl 0768.12001
Roman, Steven (1995), Field Theory, GTM 158, Springer-Verlag, ISBN 9780387944081
Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 9780130878687

[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

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