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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.
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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.
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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>
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<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>
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<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
How to Cite This Entry:
Algebraic extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_extension&oldid=36926