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Unformatted text preview: 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 . (That is, 3 = 1 but 6 = 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 : K R , with images K = ( K ), and R is generated by the union of the K s. The subgroup of the symmetric group S 3 which preserves K is the Galois group of K 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....
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