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Unformatted text preview: (1) 12 x = 2 in Z / 19 Z . (2) 7 x = 2 in Z / 24 Z . (3) 31 x = 1 in Z / 50 Z . (4) 34 x = 1 in Z / 97 Z . (5) 27 x = 2 in Z / 40 Z . (6) 15 x = 5 in Z / 63 Z . 4. (1) Let p > 2 be a prime. Prove that an equation of the form ax 2 + bx + c (where a,b,c ∈ F p ,a 6 = 0) has a solution in Z /p Z if and only if b 24 ac is a square in Z /p Z . If this is so, prove that the solutions are given by the familiar formula. (2) Determine for which values of a the equation x 2 + x + a has a solution in Z / 7 Z . 5. In each case, divide f ( x ) by g ( x ) with residue: (1) f ( x ) = 3 x 42 x 3 + 6 x 2x + 2, g ( x ) = x 2 + x + 1 in Q [ x ]. (2) f ( x ) = x 47 x + 1, g ( x ) = 2 x 2 + 1 in Q [ x ]. (3) f ( x ) = 2 x 4 + x 2x + 1, g ( x ) = 2 x1 in Z / 5 Z [ x ]. (4) f ( x ) = 4 x 4 + 2 x 3 + 6 x 2 + 4 x + 5, g ( x ) = 3 x 2 + 2 in Z / 7 Z [ x ]....
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This note was uploaded on 04/18/2008 for the course MATH 235 taught by Professor Goren during the Fall '07 term at McGill.
 Fall '07
 Goren
 Math, Algebra

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