Algebraically closed field

From Wikipedia, the free encyclopedia

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

[edit] Examples

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

3x2 + 1 = 0

has no solution in real numbers, even though both of its coefficients (3 and 1) are real. The same argument proves that the field of rational numbers is not algebraically closed. Also, no finite field F is algebraically closed, because if a1, a2, …, an are the elements of F, then the polynomial

(xa1)(xa2) ··· (xan) + 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 algebraic numbers.

[edit] Equivalent properties

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

p(x) = k(xx1)(xx2) ··· (xxn).
  • 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 / (xb)n, where n is a natural number, and a and b are elements of F.

[edit] 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 nth root in F, since this is the same thing as saying that the equation xna = 0 has some root in F. However, there are fields in which every element has an nth root (for each natural number n) but which are not algebraically closed. In fact, even assuming that every polynomial of the form xna splits into linear factors is not enough to assure that the field is algebraically closed.

Every field F has an "algebraic closure", which is the smallest algebraically closed field of which F is a subfield.

[edit] References