Unformatted text preview: det .V V t / D det 2 4 3 P ˛ 2 i P ˛ 2 i P ˛ 3 i P ˛ 2 i P ˛ 3 i P ˛ 4 i 3 5 . (b) Use equations ( 7.2.1 ) as well as the fact that the ˛ i are roots of f.x/ D to compute that P ˛ 2 i D ± 2p , P i a 3 i D ± 3q , and P i a 4 i D 2p 2 . (c) Compute that ı 2 D ± 4p 3 ± 27q 2 . 7.2.8. (a) Show that x 3 ± 2 is irreducible in Q ŒxŁ . Compute its roots and also ı 2 and ı . (b) Q . 3 p 2/ ² R , so is not equal to the splitting ﬁeld E of x 3 ± 2 . The splitting ﬁeld is Q . 3 p 2;!/ , where ! D e 2±i=3 . Show that ! satisﬁes a quadratic polynomial over Q . (c) Show that x 3 ± 3x C 1 is irreducible in Q ŒxŁ . Compute its roots, and also ı 2 and ı . Hint: The quantity A is a root of unity, and the roots are twice the real part of certain roots of unity....
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 Fall '08
 EVERAGE
 Algebra, Polynomials, Quadratic equation, Elementary algebra, Degree of a polynomial, cubic polynomial

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