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Unformatted text preview: Introduction to Ring Theory. Math 228 Unless otherwise stated, homework problems are taken from Hungerford, Abstract Algebra, Second Edition. Homework 4  due February 9 2.1.11 b, d: Find all solutions of each congruence: (b) 3 x ≡ 1 (mod 7) (d) 6 x ≡ 10 (mod 15) Solution: (b) Since (3 , 7) = 1, this equation has one solution. By simple inspection (or by Euclidean algorithm) we see that 1 = 7 3 · 2. This means that x ≡  2 ≡ 5 (mod 7) is the solution. (d) Since (6 , 15) = 3 (by factorization), this equation has no solution, since 3 6  10. 2.2.8: (a) Solve the equation x 2 + x = 0 in Z 5 . (b) Solve the equation x 2 + x = 0 in Z 6 . (c) If p is prime, prove that the only solutions of x 2 + x = 0 in Z p are 0 and p 1. Solution: (a) In Z 5 : 2 + 0 = 0 3 2 + 3 = 2 1 2 + 1 = 2 4 2 + 4 = 0 2 2 + 2 = 1 Then the solutions for x 2 + x = 0 in Z 5 are x = 0 or 4. (b) In Z 6 : 2 + 0 = 0 3 2 + 3 = 0 1 2 + 1 = 2 4 2 + 4 = 2 2 2 + 2 = 0 5 2 + 5 = 0 Then the solutions for x 2 + x = 0 in Z 6 are x = 0 , 2...
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This note was uploaded on 11/03/2010 for the course MATH 26233820 taught by Professor Gieseker,d. during the Fall '10 term at UCLA.
 Fall '10
 GIESEKER,D.
 Algebra, Congruence

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