history-page34 - Unsolvability of the Quintic • The...

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Unformatted text preview: Unsolvability of the Quintic • The Theorem: There is no formula involving the four basic arithmetic operations and root extraction applied the coefficients of a quintic (or higher degree) polynomial that is guaran- teed to produce a zero of the polynomial. • What the theorem doesn’t rule out. ◦ Other sorts of formulas. ◦ Formulas that work on specific types of polynomials. ◦ Different formulas for different polynomials. (Although this also was eventually ruled out in general.) • First explicitly conjectured by Euler in 1749. • The majority opinion credits Abel with proving this in 1826. • Others (Cauchy, Ruffini) had made substantial progress before Abel. • Abel filled in gaps in previous work. Many felt (and many still feel) that Abel’s proof itself is incomplete. • Ideas of the Proof? ◦ The coefficients of a polynomial are symmetric functions of the zeros of the polynomial....
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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.

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