Algebraically closed field

In mathematics, a field $F$ is said to be algebraically closed if every polynomial in one variable of degree at least $1$, with coefficients in $F$, has a zero (root) in $F$.

Examples

As an example, the field of real numbers is not algebraically closed, because the polynomial equation

$x^{2}+1=0$

has no solution in real numbers, even though all its coefficients (1, 0 and 1) are real. The same argument proves that the field of rational numbers is not algebraically closed either. Also, no finite field $F$ is algebraically closed, because if $a_{1}$, $a_{2}$, $\ldots$,$a_{n}$ are the elements of $F$, then the polynomial

$(x-a_{1})(x-a_{2})\cdots (x-a_{n})+1\,$

has no zero in $F$. By contrast, the field of complex numbers 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 numbers.

Equivalent properties

Given a field $F$, the assertion “$F$ is algebraically closed” is equivalent to each one of the following:

• Every polynomial $p(x)$ of degree $n$ ≥ $1$, with coefficients in $F$, splits into linear factors. In other words, there are elements $k$, $x_{1}$, $x_{2}$, …, $x_{n}$ of the field $F$ such that

$p(x)=k(x-x_{1})(x-x_{2})\cdots (x-x_{n}).\,$

• The field $F$ has no proper algebraic extension.

• For each natural number $n$, every linear map from $F^{n}$ into itself has some eigenvector.

• Every rational function in one variable $x$, with coefficients in $F$, can be written as the sum of a polynomial function with rational functions of the form $a/(x-b)^{n}$, where $n$ is a natural number, and $a$ and $b$ are elements of $F$.

Other properties

If $F$ is an algebraically closed field and $n$ is a natural number, then $F$ contains all $n$^th roots of unity, because these are (by definition) the $n$ (not necessarily distinct) zeroes of the polynomial $x^{n}-1$. 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 $x^{n}-a$ splits into linear factors is not enough to assure that the field is algebraically closed.

References

• S. Lang, Algebra, Springer-Verlag, 2004, ISBN 0-387-95385-X

• B. L. van der Waerden, Algebra I, Springer-Verlag, 1991, ISBN 0-387-97424-5