View source for Fundamental theorem of algebra

== Introduction ==

The Fundamental Theorem of Algebra (FTA) is a central theorem in the field of [[Complex Analysis|complex analysis]], a branch of [[Mathematics|mathematics]]. It states that every non-constant polynomial equation in one variable with complex coefficients has at least one complex root. This theorem links algebra, which deals with equations, and analysis, which deals with limits and continuity.

[[Image:Detail-71675.jpg|thumb|center|A complex plane with a polynomial function graphed.|class=only_on_mobile]]
[[Image:Detail-71676.jpg|thumb|center|A complex plane with a polynomial function graphed.|class=only_on_desktop]]

== History ==

The Fundamental Theorem of Algebra was first proven by [[Carl Friedrich Gauss|Gauss]] in 1799. However, his initial proof was not entirely rigorous by modern standards, and he later published several more proofs, each of which was intended to address the deficiencies of the previous one. Other mathematicians, including [[Augustin-Louis Cauchy|Cauchy]], [[Jean-Robert Argand|Argand]], and [[Bernhard Riemann|Riemann]], also contributed to the development and understanding of this theorem.

== Statement of the Theorem ==

The theorem can be stated as follows: Every non-constant polynomial function of one variable with complex coefficients has at least one complex root. In other words, if ''P'' is a polynomial in one variable with complex coefficients, and ''P'' is not constant, then there exists a complex number ''z'' such that ''P''(''z'') = 0.

== Proof ==

There are many proofs of the Fundamental Theorem of Algebra, and they use a variety of techniques from different areas of mathematics. Some of these proofs involve complex analysis, others involve algebraic topology, and still others involve algebraic geometry. Each of these proofs illuminates a different aspect of the theorem and its implications.

== Implications and Applications ==

The Fundamental Theorem of Algebra has many important implications in both theoretical and applied mathematics. It implies that the complex numbers are algebraically closed, meaning that every polynomial equation with complex coefficients has a solution in the complex numbers. This property is fundamental to many areas of mathematics, including algebraic geometry, number theory, and many areas of analysis.

In applied mathematics, the Fundamental Theorem of Algebra is used in many different contexts. For example, it is used in the study of differential equations, in the analysis of algorithms in computer science, and in the design of algorithms in numerical analysis.

== See Also ==

* [[Complex Analysis]]
* [[Algebraic Topology]]
* [[Algebraic Geometry]]
* [[Number Theory]]
* [[Differential Equations]]
* [[Computer Science]]
* [[Numerical Analysis]]

[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Complex Analysis]]