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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 11/03/2010 for the course MATH 262-338-20 taught by Professor Gieseker,d. during the Fall '10 term at UCLA.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online