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Wikipedia: 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 $a 1$ , $a 2$ , …, $a n$ 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

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algebraically closed field

Examples

Equivalent properties

Other properties

References

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