The following sections of this BookRags Literature Study Guide is offprint from Gale's For Students Series: Presenting Analysis, Context, and Criticism on Commonly Studied Works: Introduction, Author Biography, Plot Summary, Characters, Themes, Style, Historical Context, Critical Overview, Criticism and Critical Essays, Media Adaptations, Topics for Further Study, Compare & Contrast, What Do I Read Next?, For Further Study, and Sources.
(c)1998-2002; (c)2002 by Gale. Gale is an imprint of The Gale Group, Inc., a division of Thomson Learning, Inc. Gale and Design and Thomson Learning are trademarks used herein under license.
The following sections, if they exist, are offprint from Beacham's Encyclopedia of Popular Fiction: "Social Concerns", "Thematic Overview", "Techniques", "Literary Precedents", "Key Questions", "Related Titles", "Adaptations", "Related Web Sites". (c)1994-2005, by Walton Beacham.
The following sections, if they exist, are offprint from Beacham's Guide to Literature for Young Adults: "About the Author", "Overview", "Setting", "Literary Qualities", "Social Sensitivity", "Topics for Discussion", "Ideas for Reports and Papers". (c)1994-2005, by Walton Beacham.
All other sections in this Literature Study Guide are owned and copyrighted by BookRags, Inc.
The fundamental theorem of algebra is the statement that every polynomial with complex numbers as coefficients has a complex number as a root, or equivalently, that such a polynomial has n roots, where n is its degree, in the complex numbers, counted with multiplicitly (that is, double roots count double and so on).
While this can be checked for polynomials of degree up to four, using the formulas for the roots, this approach will not work for higher degrees. Some early authors seem to have taken the fundamental theorem of algebra for granted, without realizing it required a justification. Albert Girard (1629) might have been the first to call attention to the statement of the fundamental theorem of algebra but without trying to justify it. The famous mathematician and philosopher Gotthold Leibniz (1702) even doubted its validity. The first serious attempt at a proof was made by the French mathematician Jean Le Rond D'Alembert in 1746 but his proof was incomplete. The first correct proof was given by the great German mathematician Carl Frederich Gauss in 1799. Gauss then subsequently gave three other proofs and since then there has been many more different proofs. Stricly speaking, the fundamental theorem of algebra is really a theorem in Analysis, since its truth rests on the continuity properties of real and complex numbers. The algebraic content of the theorem has been made explicit by the theory of real closed fields developed by Emil Artin and Otto Schreier (1926).