Fundamental theorem of algebra (idea) by neil

Every complex polynomial (including real polynomials) has at least one complex root. Actually, each complex polynomial of degree n has n roots, but some of those roots may be the same.

A more general concept is algebraic closure. A field F is algebraically closed if every nonconstant polynomial over F splits over F. The FTA says that C is algebraically closed. For more information on polynomials, splitting fields, and algebraic closure, consult your favourite algebra book's chapter on field extensions. For a proof of the FTA, consult any book on complex analysis---the proof is not very complicated, but it relies on more than just algebra.

Absolute Terror Field	Fundamental theorem of calculus	holomorphic function	Laguerre's method
Liouville's Theorem	QED	Node More Mathematics	Effects on an unprotected human body in the vacuum of space
mathematics	Proof that the algebraic closure of a field is unique	The Punch Thyself	algebraically closed
splitting field	Hairy ball theorem	Galois Theory	root
Wilkinson's polynomial	Borsuk-Ulam Theorem	Fundamental theorem of arithmetic	algebra
The abc conjecture	nth roots of unity	division rule for seven	Is it possible to pack two transcendental numbers?