less than 1 minute read

Fundamental Theorems

Fundamental Theorem Of Algebra



The fundamental theorem of algebra asserts that every polynomial equation of degree n ≥ 1, with complex coefficients, has at least one solution among the complete numbers. An important result of this theorem says that the set of complex numbers is algebraically closed; meaning that if the coefficients of every polynomial equation of degree n are contained in a given set, then every solution of every such polynomial equation is also contained in that set. To see that the set of real numbers is not algebraically closed consider the origin of the imaginary number i. Historically, i was invented to provide a solution to the equation x2 + 1 = 0, which is a polynomial equation of degree 2 with real coefficients. Since the solution to this equation is not a real number, the set of real numbers is not algebraically closed. That the complex numbers are algebraically closed, is of basic or fundamental importance to algebra and the solution of polynomial equations. It implies that no polynomial equation exists that would require the invention of yet another set of numbers to solve it.




Additional topics

Science EncyclopediaScience & Philosophy: Formate to GastropodaFundamental Theorems - Fundamental Theorem Of Algebra, Fundamental Theorem Of Calculus - Fundamental theorem of arithmetic