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Algebraic extension

From Encyclopedia of Mathematics
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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 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.

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