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 $F$.

Contents

Examples

Equivalent properties

Other properties

References

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 $a\_1$, $a\_2$, …, $a\_n$ 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, $a$ is an element of $F$, and $n$ is a natural number, then $a$ has an $n$^th root in $F$, since this is the same thing as saying that the equation $x^n-a=0$ has some root in $F$. However, there are fields in which every element has an $n$^th root (for each natural number $n$) but which are not algebraically closed. In fact, 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.
Assuming Zorn's lemma, every field $F$ has a unique algebraic closure, which is the smallest algebraically closed field of which $F$ is a subfield.

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