In [[mathematics]], a [[field (mathematics)|field]] <math>F</math> is said to be '''algebraically closed''' if every [[polynomial]] in one [[variable]] of degree at least <math>1</math>, with [[coefficient]]s in <math>F</math>, has a [[root (mathematics)|zero]] ([[root (mathematics)|root]]) in <math>F</math>.
==Examples==
As an example, the field of [[real number]]s is not algebraically closed, because the polynomial equation <math>x^2+1=0</math> has no solution in real numbers, even though all its coefficients (1, 0 and 1) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of [[rational number]]s is not algebraically closed. Also, no [[finite field]] <math>F</math> is algebraically closed, because if <math>a_1 </math>, <math> a_2 </math>, …, <math> a_n</math> are the elements of <math>F</math>, then the polynomial
:<math>(x-a_1)(x-a_2)\cdots(x-a_n)+1\,</math>
has no zero in <math>F</math>. By contrast, the field of [[complex number]]s is algebraically closed: this is stated by the [[fundamental theorem of algebra]]. Another example of an algebraically closed field is the field of (complex) [[algebraic number]]s.
==Equivalent properties==
Given a field <math>F</math>, the assertion “<math>F</math> is algebraically closed” is equivalent to each one of the following:
* The only [[irreducible polynomial|irreducible polynomials]] in the [[Ring (mathematics)|ring]] <math>F[x]</math> are those of degree one.
* Every polynomial <math>p(x)</math> of degree <math>n</math> ≥ <math>1</math>, with [[coefficient]]s in <math>F</math>, [[factorization|splits into linear factors]]. In other words, there are elements <math>k</math>, <math>x_1</math>, <math>x_2</math>, …, <math>x_n</math> of the field <math>F</math> such that
::<math>p(x)=k(x-x_1)(x-x_2) \cdots (x-x_n).\,</math>
* The field <math>F</math> has no proper [[algebraic extension]].
* For each natural number <math>n</math>, every [[linear map]] from <math>F^n</math> into itself has some [[eigenvector]].
* Every [[rational function]] in one variable <math>x</math>, with coefficients in <math>F</math>, can be written as the sum of a polynomial function with rational functions of the form <math>a/(x-b)^n</math>, where <math>n</math> is a natural number, and <math>a</math> and <math>b</math> are elements of <math>F</math>.
==Other properties==
If <math>F</math> is an algebraically closed field and <math>n</math> is a natural number, then <math>F</math> contains all <math>n</math><sup>th</sup> roots of unity, because these are (by definition) the <math>n</math> (not necessarily distinct) zeroes of the polynomial <math>x^n-1</math>. A field extension that is contained in an extension generated by the roots of unity is a ''cyclotomic extension'', and the extension of a field generated by all roots of unity is sometimes called its ''cyclotomic closure''. Thus algebraically closed fields are cyclotomically closed. The converse is not true. Even assuming that every polynomial of the form <math>x^n-a</math> splits into linear factors is not enough to assure that the field is algebraically closed.
If a proposition which can be expressed in the language of [[First-order logic|first-order logic]] is true for an algebraically closed field, then it is true for every algebraically closed field with the same [[Characteristic|characteristic]]. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only it is valid for all other algebraically closed field with characteristic 0 as there is some natural number <math>N</math> such that the proposition is valid for every algebraically closed field with characteristic <math>p</math> when <math>p>N</math>.<ref>See subsections ''Rings and fields'' and ''Properties of mathematical theories'' in §2 of J. Barwise's "An introduction to first-order logic".</ref>
Every field <math>F</math> has some extension which is algebraically closed. Among all such extensions there is one and ([[Up to|up to isomorphism]]) only one which is an [[algebraic extension]] of <math>F</math>;<ref>See Lang's ''Algebra'', §VII.2 or van der Waerden's ''Algebra I'', §10.1.</ref> it is called the [[algebraic closure]] of <math>F</math>.
==Notes==
{{reflist}}
==References==
* {{citation | last = Barwise | first = Jon | author-link = Jon Barwise | year = 1978 | contribution = An introduction to first-order logic | editor-last = Barwise | editor-first = Jon | title = Handbook of Mathematical Logic | series = Studies in Logic and the Foundations of Mathematics | publisher = North Holland | isbn = 0-7204-2285-X}}
* {{citation | last = Lang | first = Serge | author-link = Serge Lang | title = Algebra | edition = 4th | date = 2004 | series = Graduate Texts in Mathematics 211 | publisher = Springer-Verlag | isbn = 0-387-95385-X}}
* {{citation | last = van der Waerden | first = Bartel Leendert | author-link = Bartel Leendert van der Waerden | title = Algebra | volume = I | edition = 7th | date = 2003 | publisher = Springer-Verlag | isbn = 0-387-40624-7}}
[[Category:Abstract algebra]]
[[Category:Field theory]]
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