Difference between revisions of "Algebraic extension"
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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 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. | 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. | ||
| − | Algebraic extensions form a ''distinguished class'' : that is, they have the properties (i) for $M / L / K$ we have $M/L,\,L/K$ algebraic if and only if $M/K$ is algebraic; (ii) $M / K,\,L/K $ algebraic implies $ ML/L$ algebraic. | + | Algebraic extensions form a ''[[Distinguished class of extensions|distinguished class]]'' : that is, they have the properties (i) for $M / L / K$ we have $M/L,\,L/K$ algebraic if and only if $M/K$ is algebraic; (ii) $M / K,\,L/K $ algebraic implies $ ML/L$ algebraic. |
An extension which is not algebraic is a [[transcendental extension]]. | An extension which is not algebraic is a [[transcendental extension]]. | ||
Revision as of 18:26, 18 October 2017
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.
Algebraic extensions form a distinguished class : that is, they have the properties (i) for $M / L / K$ we have $M/L,\,L/K$ algebraic if and only if $M/K$ is algebraic; (ii) $M / K,\,L/K $ algebraic implies $ ML/L$ algebraic.
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 Zbl 0768.12001 |
| [b2] | Steven Roman, Field Theory, Graduate Texts in Mathematics 158 (2nd edition) Springer (2007) ISBN 0-387-27678-5 Zbl 1172.12001 |
Algebraic extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_extension&oldid=42117