Difference between revisions of "Abel-Ruffini theorem"

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The '''Abel-Ruffini theorem''', often known just as '''Abel's theorem''', states that it is impossible to solve a general quintic equation "in radicals".  This is in stark contrast with [[polynomial|polynomials]] of smaller [[degree]].
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Recall that a "quadratic polynomial" is an expression of the form <math>p(x) = ax^2+bx+c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are some real (or complex) coefficients.  It is a familiar fact that the solutions of <math>ax^2+bx+c</math> are given by the [[quadratic formula]] as
 
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<math>x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}.</math>
 
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It was long wondered whether it was possible to give a similar formula for polynomials of larger degree, for example cubics <math>ax^3+bx^2+cx+d=0</math> and quartics <math>ax^4+bx^3+cx^2+dx+e=0</math>.  The explicit, but more elaborate, "cubic formula" and "quartic formula" were derived by Cardano and Ferrari by the 16th century.  The first of these involves taking a number of cube roots, and the later a number of fourth roots.  It was long wondered whether a similar formula existed for quintic polynomials, so that solutions could be computed through some complex formula involving fifth roots, but Abel's theorem states that this is impossible: there is no way to solve a general quintic "in radicals".  A standard example of a quintic whose roots may not be expressed as radicals is <math>x^5-x+1=0</math>.
 
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The Abel-Ruffini theorem provided the impetus for the development of the modern field of [[Galois]] theory, and indeed much of [[abstract algebra]].  Once some amount of Galois theory is understood, the impossibility of solving a general quintic turns out to be a consequence of some easy facts in [[group theory]]: in particular, the fact that the permutation group <math>S_5</math> is not "solvable".
 
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[[Category:Mathematics]]
 

Revision as of 14:26, September 13, 2011

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