Difference between revisions of "Algebraically closed field"
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| − | 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|Algebra, fundamental theorem of]]). | + | 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|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|Algebra, fundamental theorem of]]). | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Algebra", Addison-Wesley (1974) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|ZaSa}}||valign="top"| O. Zariski, P. Samuel, "Commutative algebra", '''1''', Springer (1975) | ||
| + | |- | ||
| + | |} | ||
Revision as of 18:13, 17 February 2012
2020 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) |
| [ZaSa] | O. Zariski, P. Samuel, "Commutative algebra", 1, Springer (1975) |
How to Cite This Entry:
Algebraically closed field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraically_closed_field&oldid=11228
Algebraically closed field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraically_closed_field&oldid=11228
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article