cubics - Math 220C April 25, 2008 Solving the cubic...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 08/06/2008 for the course MATH 220 taught by Professor Morrison during the Spring '08 term at UCSB.

Page1 / 2

cubics - Math 220C April 25, 2008 Solving the cubic...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online