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Algebraic extension

Inmathematics, an algebraic extension is a field extension L/K such that every element of the larger field Lisalgebraic 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 ${\displaystyle \mathbb {Q} }$ 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 ${\displaystyle \mathbb {C} /\mathbb {R} }$ of the real numbers by the complex numbers.

Some properties[edit]

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 ∈ E. The smallest subfield of E that contains K and a is commonly denoted ${\displaystyle K($a).} Ifa 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, ${\displaystyle K($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, ${\displaystyle \mathbb {Q} (\pi )\neq \mathbb {Q} [\pi ],}$ and they are both infinite dimensional vector spaces over ${\displaystyle \mathbb {Q} .}$^[9]

Analgebraically 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-algebraofL is a field.

Properties[edit]

The following three properties hold:^[12]

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

4. 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[edit]

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embeddingofM into N is called an algebraic extension if for every xinN there is a formula p with parameters in M, such that p(x) is true and the set

${\displaystyle \left\{y\in N\mid p($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 groupofN over M can again be defined as the groupofautomorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case.

Relative algebraic closures[edit]

Given a field k and a field K containing k, one defines the relative algebraic closureofkinK 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[edit]

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

Notes[edit]

References[edit]

• 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