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Unformatted text preview: f.x/ D x 3 C px C q has no quadratic term. If f has a double root 1 D 2 D , then the third root is 3 D 2 . Now, observe that the relation X i<j i j D p shows that satises a quadratic polynomial over K . 7.4.2. Expand the expression D . 1 2 /. 1 3 /. 2 3 / , and reduce the result using Equations ( 7.4.1 ) through ( 7.4.3 ) to eliminate 3 and to reduce higher powers of 1 and 2 . Show that D 6 2 1 2 2 1 p C 2 2 p 3q . Solve for 2 to get 2 D C 2 1 p C 3q 2 3 1 2 C p : Explain why the denominator in this expression is not zero. 7.4.3. (a) Conrm the details of Example 7.4.10 . (b) Consider the irreducible polynomial x 3 3x C 1 in Q x . Show that the dimension over Q of the splitting eld is 3. Conclude that there are no elds intermediate between Q and the splitting eld....
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 Fall '08
 EVERAGE
 Algebra

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