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ass6 - ASSIGNMENT 6 MATH235 FALL 2007 Submit by 16:00...

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ASSIGNMENT 6 - MATH235, FALL 2007 Submit by 16:00, Monday, October 29 1. Calculate the following: (1) (2 19808 + 6) - 1 + 1 (mod 11). (2) 12 , 12 2 , 12 4 , 12 8 , 12 16 , 12 25 all modulo 29. (Hint: think before computing). 2. Use the Euclidean algorithm to find the gcd of the following pairs of polynomials and express it as a combination of the two polynomials. (1) x 4 - x 3 - x 2 + 1 and x 3 - 1 in Q [ x ]. (2) x 5 + x 4 + 2 x 3 - x 2 - x - 2 and x 4 + 2 x 3 + 5 x 2 + 4 x + 4 in Q [ x ]. (3) x 4 + 3 x 3 + 2 x + 4 and x 2 - 1 in Z / 5 Z [ x ]. (4) 4 x 4 + 2 x 3 + 3 x 2 + 4 x + 5 and 3 x 3 + 5 x 2 + 6 x in Z / 7 Z [ x ]. (5) x 3 - ix 2 + 4 x - 4 i and x 2 + 1 in C [ x ]. (6) x 4 + x + 1 and x 2 + x + 1 in Z / 2 Z [ x ]. 3. Consider the polynomial x 2 + x = 0 over Z /n Z . (1) Find an n such that the equation has at least 4 solutions. (2) Find an n such that the equation has at least 8 solutions. 4. Is the given polynomial irreducible: (1) x 2 - 3 in Q [ x ]? In R [ x ]? (2) x 2 + x - 2 in F 3 [ x ]? In F 7 [ x ]? 5. Find the rational roots of the polynomial 2 x 4 + 4 x 3 - 5 x 2 - 5 x + 2. 6. Recall that for the ring Z a complete list of ideals is given by (0) , (1) , (2) , (3) , (4) , (5) , . . . , where ( n ) is the principal ideal generated by n , namely, ( n ) = { na : a Z } . Find the complete list of ideals of the
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