College Algebra Exam Review 307

College Algebra Exam Review 307 - Chapter 7 Field...

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Chapter 7 Field Extensions – First Look 7.1. A Brief History The most traditional concern of algebra is the solution of polynomial equa- tions and related matters such as computation with radicals. Methods of solving linear and quadratic equations were known to the ancients, and Arabic scholars preserved and augmented this knowledge during the Mid- dle Ages. You learned the quadratic formula for the roots of a quadratic equation in school, but more fundamental is the algorithm of completing the square, which justifies the formula. Similar procedures for equations of the third and fourth degree were discovered by several mathematicians in sixteenth century Italy, who also introduced complex numbers (with some misgiv- ings). And there the matter stood, more or less, for another 250 years. At the end of the eighteenth century, no general method or formula, of the sort that had worked for equations of lower degree, was known for equations of degree 5, nor was it known that no such method was possible. The sort of method sought was one of “solution by radicals,” that is, by
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