hwk228-4 - Introduction to Ring Theory Math 228 Unless...

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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 : 0 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 : 0 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 , 3 , or 5. (c) x 2 + x 0 (mod p ) if and only if p | ( x 2 + x ) = x ( x + 1). Since p is prime, this happens if and only if p | x or p | ( x +1). In the first case, x 0 (mod
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