College Algebra Exam Review 325

College Algebra Exam Review 325 - f.x D x 3 C px C q has no...

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7.4. SPLITTING FIELD OF A CUBIC POLYNOMIAL 335 Example 7.4.10. Consider f.x/ D x 3 ± 2 , which is irreducible over Q . The three roots of f in C are 3 p 2 , ! 3 p 2 , and ! 2 3 p 2 , where ! D ± 1=2 C p ± 3=2 is a primitive cube root of 1. Let E denote the splitting field of f in C . The discriminant of f is ı 2 D ± 108 , which has square root ı D 6 p ± 3 . It follows that the Galois group G D Aut Q .E/ is of order 6, and dim Q .E/ D 6 . Each of the fields Q .˛/ , where ˛ is one of the roots of f , is a cubic extension of Q , and is the fixed field of the order 2 subgroup of G that exchanges the other two roots. The only other intermediate field between Q and E is Q .ı/ D Q .!/ D Q . p ± 3/ , which is the fixed field of the alternating group A 3 Š Z 3 . A diagram of intermediate fields, with the dimensions of the field extensions indicated is shown as Figure 7.4.1 . Exercises 7.4 7.4.1. Let K ² C be a field. (a) Suppose f.x/ 2 KŒxŁ is an irreducible quadratic polynomial. Show that the two roots of f.x/ in C are distinct. (b) Suppose f.x/ 2 KŒxŁ is an irreducible cubic polynomial. Show that the three roots of f.x/ in C are distinct. Hint: We can assume without loss of generality that
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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 ˛ 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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