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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.21 The Insolubility of the Quintic

Martin W. Liebeck

Every student will be familiar with the formula for the roots of a quadratic polynomial ax2 + bx + c, namely (-b ± Image)/2a. Perhaps less familiar is the fact that there is also a formula for the roots of a cubic: write the cubic as x3 + ax2+ bx+c, and make the substitution y = x + Imagea to rewrite it in the form y3 + 3hy + k. The roots of this are then of the form

Image

While the quadratic formula was known ...

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