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cubics - Math 220C Solving the cubic equation We want to...

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Math 220C April 25, 2008 Solving the cubic equation We want to find the roots of the general cubic polynomial f ( x ) = x 3 + ax 2 + bx + c F [ x ] , where F is a field containing a nontrivial cube root of unity ζ . (That is, ζ 3 = 1 but ζ = 1.) We also assume that the characteristic of F is not 2. We work in the ring R = F [ r 1 , r 2 , r 3 ] / ( a + r 1 + r 2 + r 3 , b - r 1 r 2 - r 2 r 3 - r 3 r 1 , c + r 1 r 2 r 3 ) . If K is any splitting field for f ( x ) over F , then there are 3! monomorphisms ψ σ : K R , with images K σ = ψ σ ( K ), and R is generated by the union of the K σ ’s. The subgroup of the symmetric group S 3 which preserves K σ is the Galois group of K σ over F . Our analysis of the roots will use the action of S 3 on R ; at the end, we will consider the issue of determining the Galois group of the splitting field itself. The formulas we will derive are much cleaner if we first make a change of variables x = y - a 3 . This has the effect of replacing the original polynomial f ( x ) by g ( y ) = y 3 + b - 1 3 a 2 y + 2 27 a 3 - 1 3 ab + c = y 3 + py + q, where p = b - 1 3 a 2 , q =
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