Difference between revisions of "Algebraic extension"
From Encyclopedia of Mathematics
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| − | A [[field extension]] $K/k$ in which every element of $K$ is algebraic over $k$; that is, every element of $K$ is the root of a polynomial with coefficients in $k$. A finite degree extension is necessarily algebraic, but the converse does not hold: for example, the field of [[algebraic number]]s, the [[algebraic closure]] of the field of [[rational number]]s, is an algebraic extension but not of finite degree. | + | A [[field extension]] $K/k$ in which every element of $K$ is algebraic over $k$; that is, every element of $K$ is the root of a non-zero polynomial with coefficients in $k$. A finite degree extension is necessarily algebraic, but the converse does not hold: for example, the field of [[algebraic number]]s, the [[algebraic closure]] of the field of [[rational number]]s, is an algebraic extension but not of finite degree. |
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| + | An extension which is not algebraic is a [[transcendental extension]]. | ||
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Revision as of 19:42, 13 December 2015
2020 Mathematics Subject Classification: Primary: 12F [MSN][ZBL]
A field extension $K/k$ in which every element of $K$ is algebraic over $k$; that is, every element of $K$ is the root of a non-zero polynomial with coefficients in $k$. A finite degree extension is necessarily algebraic, but the converse does not hold: for example, the field of algebraic numbers, the algebraic closure of the field of rational numbers, is an algebraic extension but not of finite degree.
An extension which is not algebraic is a transcendental extension.
References
| [b1] | Paul J. McCarthy, "Algebraic Extensions of Fields", Courier Dover Publications (2014) ISBN 048678147X |
How to Cite This Entry:
Algebraic extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_extension&oldid=36930
Algebraic extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_extension&oldid=36930