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 aE. The smallest subfield of E that contains K and a is commonly denoted 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, 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, and they are both infinite dimensional vector spaces over [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.

Relative algebraic closures[edit]

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.

Integral element

Lüroth's theorem

Galois extension

Separable extension

Normal extension

Fraleigh, John B. (2014), A First Course in Abstract Algebra, Pearson,  978-1-292-02496-7

ISBN

; Gubareni, Nadiya; Gubareni, Nadezhda Mikhaĭlovna; Kirichenko, Vladimir V. (2004), Algebras, rings and modules, vol. 1, Springer, ISBN 1-4020-2690-0

Hazewinkel, Michiel

(1993), "V.1:Algebraic Extensions", Algebra (Third ed.), Reading, Mass.: Addison-Wesley, pp. 223ff, ISBN 978-0-201-55540-0, Zbl 0848.13001

Lang, Serge

Malik, D. B.; Mordeson, John N.; Sen, M. K. (1997), Fundamentals of Abstract Algebra, McGraw-Hill,  0-07-040035-0

ISBN

McCarthy, Paul J. (1991) [corrected reprint of 2nd edition, 1976], , New York: Dover Publications, ISBN 0-486-66651-4, Zbl 0768.12001

Algebraic extensions of fields

Roman, Steven (1995), , GTM 158, Springer-Verlag, ISBN 9780387944081

Field Theory

Rotman, Joseph J. (2002), , Prentice Hall, ISBN 9780130878687

Advanced Modern Algebra