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 ˛ satisﬁes a quadratic polynomial over K . 7.4.2. Expand the expression ı 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 ı D 6˛ 2 1 ˛ 2 ± 2˛ 1 p C 2˛ 2 p ± 3q . Solve for ˛ 2 to get ˛ 2 D ı C 2˛ 1 p C 3q 2 ± 3˛ 1 2 C p ² : Explain why the denominator in this expression is not zero. 7.4.3. (a) Conﬁrm the details of Example 7.4.10 . (b) Consider the irreducible polynomial x 3 ± 3x C 1 in Q ŒxŁ . Show that the dimension over Q of the splitting ﬁeld is 3. Conclude that there are no ﬁelds intermediate between Q and the splitting ﬁeld....
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 Fall '08
 EVERAGE
 Algebra, Polynomials, Quadratic equation, Elementary algebra, Complex number

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