Inmathematics, analgebraic extension is afield extensionL/K such that every element of the largerfieldL isalgebraic over the smaller fieldK; that is, every element ofL is a root of a non-zeropolynomial with coefficients inK.^[1][2] A field extension that is not algebraic, is said to betranscendental, and must containtranscendental elements, that is, elements that are not algebraic.^[3][4]

The algebraic extensions of the field${\displaystyle \mathbb{Q}}$ of therational numbers are calledalgebraic number fields and are the main objects of study ofalgebraic number theory. Another example of a common algebraic extension is the extension${\displaystyle \mathbb{C}/\mathbb{R}}$ of thereal numbers by thecomplex numbers.

All transcendental extensions are of infinitedegree. This in turn implies that all finite extensions are algebraic.^[5] Theconverse is not true however: there are infinite extensions which are algebraic.^[6] For instance, the field of allalgebraic numbers is an infinite algebraic extension of the rational numbers.^[7]

LetE be an extension field ofK, anda ∈E. The smallest subfield ofE that containsK anda is commonly denoted${\displaystyle K(a).}$ Ifa is algebraic overK, then the elements ofK(a) can be expressed as polynomials ina with coefficients inK; that is,${\displaystyle K(a)=K[a]}$, the smallestring containingK anda. In this case,${\displaystyle K(a)}$ is a finite extension ofK and all its elements are algebraic overK. In particular,${\displaystyle K(a)}$ is aK-vector space with basis${\displaystyle \{1,a,...,a^{d-1}\}}$, whered is the degree of theminimal polynomial ofa.^[8] These properties do not hold ifa is not algebraic. For example,${\displaystyle \mathbb{Q}(\pi )\ne \mathbb{Q}[\pi ],}$ and they are both infinite dimensional vector spaces over${\displaystyle \mathbb{Q}.}$^[9]

Analgebraically closed fieldF has no proper algebraic extensions, that is, no algebraic extensionsE withF <E.^[10] An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called itsalgebraic closure), butproving this in general requires some form of theaxiom of choice.^[11]

An extensionL/K is algebraicif and only if every subK-algebra ofL is a field.

The following three properties hold:^[12]

1. IfE is an algebraic extension ofF andF is an algebraic extension ofK thenE is an algebraic extension ofK.
2. IfE andF are algebraic extensions ofK in a common overfieldC, then thecompositumEF is an algebraic extension ofK.
3. IfE is an algebraic extension ofF andE >K >F thenE is an algebraic extension ofK.

These finitary results can be generalized using transfinite induction:

This fact, together withZorn's lemma (applied to an appropriately chosenposet), establishes the existence ofalgebraic closures.

Model theory generalizes the notion of algebraic extension to arbitrary theories: anembedding ofM intoN is called analgebraic extension if for everyx inN there is aformulap with parameters inM, such thatp(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. TheGalois group ofN overM can again be defined as thegroup ofautomorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case.

Given a fieldk and a fieldK containingk, one defines therelative algebraic closure ofk inK to be the subfield ofK consisting of all elements ofK that are algebraic overk, that is all elements ofK that are a root of some nonzero polynomial with coefficients ink.

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

• 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

• Field extensions

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