Algebraically closed field
A field in which any polynomial of non-zero degree over
has at least one root. In fact, it follows that for an algebraically closed field
each polynomial of degree
over
has exactly
roots in
, i.e. each irreducible polynomial from the ring of polynomials
is of degree one. A field
is algebraically closed if and only if it has no proper algebraic extension (cf. Extension of a field). For any field
, there exists a unique (up to isomorphism) algebraic extension of
that is algebraically closed; it is called the algebraic closure of
and is usually denoted by
. Any algebraically closed field containing
contains a subfield isomorphic to
.
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
[1] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) |
[2] | S. Lang, "Algebra" , Addison-Wesley (1974) |
Algebraically closed field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraically_closed_field&oldid=11228