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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 2 D 4p 3 27q 2 . 7.2.8. (a) Show that x 3 2 is irreducible in Q x . Compute its roots and also 2 and . (b) Q . 3 p 2/ R , so is not equal to the splitting eld E of x 3 2 . The splitting eld is Q . 3 p 2;!/ , where ! D e 2i=3 . Show that ! satises a quadratic polynomial over Q . (c) Show that x 3 3x C 1 is irreducible in Q x . Compute its roots, and also 2 and . 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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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08