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 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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| + | 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. | ||
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| + | An extension which is not algebraic is a [[transcendental extension]]. | ||
====References==== | ====References==== | ||
<table> | <table> | ||
| − | <TR><TD valign="top">[b1]</TD> <TD valign="top"> Paul J. McCarthy, "Algebraic Extensions of Fields", Courier Dover Publications (2014) ISBN 048678147X </TD></TR> | + | <TR><TD valign="top">[b1]</TD> <TD valign="top"> Paul J. McCarthy, "Algebraic Extensions of Fields", Courier Dover Publications (2014) {{ISBN|048678147X}} {{ZBL|0768.12001}} </TD></TR> |
| + | <TR><TD valign="top">[b2]</TD> <TD valign="top"> Steven Roman, ''Field Theory'', Graduate Texts in Mathematics '''158''' (2nd edition) Springer (2007) {{ISBN|0-387-27678-5}} {{ZBL|1172.12001}}</TD></TR> | ||
</table> | </table> | ||
Latest revision as of 14:18, 12 November 2023
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=36926