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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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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Radicals, Equations

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