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The fundamental theorem of algebra

In mathematics, the fundamental theorem of algebra states that every complex polynomial [math]p(z)[/math] in one variable and of degree [math]n[/math] ≥ [math]1[/math] has some complex root. In other words, the field of complex numbers is algebraically closed and therefore, as for any other algebraically closed field, the equation [math]p(z)=0[/math] has [math]n[/math] roots (not necessarily distinct).

The name of the theorem is now considered a misnomer by many mathematicians, since it is an instance of analysis rather than algebra.

Algebraically closed field

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

What is a field?

In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.

The mathematical discipline concerned with the study of fields is called field theory.

Abstract algebra and abstract structures

Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, w:vector spaces, and algebras. Most authors nowadays simply write algebra instead of abstract algebra.

The term abstract algebra now refers to the study of all algebraic structures, as distinct from the elementary algebra taught in schools, which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers, and unknowns. Elementary algebra can be taken as an informal introduction to the structures known as the real field and commutative algebra.

Contemporary mathematics and mathematical physics make intensive use of abstract algebra; for example, theoretical physics draws on Lie algebras. Fields such as algebraic number theory, algebraic topology, and algebraic geometry apply algebraic methods to other areas of mathematics. Representation theory, roughly speaking, takes the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure; see model theory.

Two mathematical fields that study the properties of algebraic structures viewed as a whole are universal algebra and category theory. Algebraic structures, together with the associated homomorphisms, form categories. Category theory is a powerful formalism for studying and comparing different algebraic structures.

In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties.

Abstractly, an "algebraic structure," is the collection of all possible models of a given set of axioms. More concretely, an algebraic structure is any particular model of some set of axioms. For example, the monster group both "is" an algebraic structure in the concrete sense, and abstractly, "has" the group structure in common with all other groups. This article employs both meanings of "structure."

This definition of an algebraic structure should not be taken as restrictive. Anything that satisfies the axioms defining a structure is an instance of that structure, regardless of how many other axioms that instance happens to have. For example, all groups are also semigroups and magmas.