The Fundamental Theorem of Algebra - Fine, Benjamin 1948- (Springer New York, 1997)
These notes grew outoftwo courses, one given in the United States and one given in Germany on the Fundamental Theorem of Algebra. The purpose ofthese courses was to present a great deal ofnonelementary mathematics, all centered on a single topic. The Fundamental Theorem ofAlgebrawasideal forthispurpose. Analysis, algebraandtopologyeach have developed different techniques which surround this result. These techniques lead to different proofs and different views of this impor tant result. Itis startlinghow much mathematics canbe introduced and learned in this manner. Inthe United States it was presented as a "capstone" course for upper level undergraduates. Many ofthe topics were familiar to the students but many were new. The goal ofcontinually returning to a proofofthe FundamentalTheoremofAlgebragave a focus to alargebodyof(whatis at first glance) seeminglyunrelated material. In addition, many nice ap plications, suchas the insolvabilityofthe quinticand the transcendence of e and 7T: could be introduced. We feel that undergraduates in such a capstone course are an ideal audience for the book. Many departments in the U.S. are adopting the idea ofa summary course. In addition, the book could serve as a foundation reference for beginning graduate stu dents. We also feel that the algebra sections, Chapters2,3,6, 7, couldbe used, withsomeadditions from outsidesources, asanalternativeversion ofanundergraduatealgebracourseorasa supplementfor suchacourse. The United States version ofthe course covered in one semester, with someomissions, mostofthe materialinChapters 1through 7. The whole bookcouldbe covered at a relatively moderate pace in two semesters. viii Preface InGermanythematerialwaspresentedtoaclassofpotentialteachers