View source for Algebraically closed field

{{MSC|12Exx|12Fxx}}
{{TEX|done}}


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====
{|
|-
|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Algebra", Addison-Wesley (1974)  {{MR|0783636}} {{ZBL|0712.00001}}
|-
|valign="top"|{{Ref|ZaSa}}||valign="top"| O. Zariski, P. Samuel, "Commutative algebra", '''1''', Springer (1975) {{MR|0384768}} {{ZBL|0313.13001}}
|-
|}