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Algebraically closed field

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2010 Mathematics Subject Classification: Primary: 12Exx Secondary: 12Fxx [MSN][ZBL]


A field $k$ is algebraically closed if any polynomial of non-zero degree over $k$ has at least one root in $k$. In fact, it follows that for an algebraically closed field $k$ each polynomial of degree $n$ over $k$ has exactly $n$ roots in $k$, i.e. each irreducible polynomial from the ring of polynomials $k[x]$ is of degree one. A field $k$ is algebraically closed if and only if it has no proper algebraic extension (cf. Extension of a field). For any field $k$, there exists a unique (up to isomorphism) algebraic extension of $k$ that is algebraically closed; it is called the algebraic closure of $k$ and is usually denoted by $\bar k$. Any algebraically closed field containing $k$ contains a subfield isomorphic to $k$.

The field of complex numbers is the algebraic closure of the field of real numbers. This is the fundamental theorem of algebra (cf. Algebra, fundamental theorem of).

References

[La] S. Lang, "Algebra", Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[ZaSa] O. Zariski, P. Samuel, "Commutative algebra", 1, Springer (1975) MR0384768 Zbl 0313.13001
How to Cite This Entry:
Algebraically closed field. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Algebraically_closed_field&oldid=21550
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article