College Algebra Exam Review 34

College Algebra Exam Review 34 - D 0 in Z n . 1.7.14....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
44 1. ALGEBRAIC THEMES (for n ± 10 ) that every nonzero element in Z n is either invertible or a zero divisor? 1.7.10. Based on your data for Z n with n ± 10 , make a conjecture (guess) about which elements in Z n are invertible and which are zero divisors. Does your conjecture imply that every nonzero element is either invertible or a zero divisor? The next three exercises provide a guide to a more analytical approach to invertibility and zero divisors in Z n . 1.7.11. Suppose a is relatively prime to n . Then there exist integers s and t such that as C nt D 1 . What does this say about the invertibility of ŒaŁ in Z n ? 1.7.12. Suppose a is not relatively prime to n . Then there do not exist integers s and t such that as C nt D 1 . What does this say about the invertibility of ŒaŁ in Z n ? 1.7.13. Suppose that ŒaŁ is not invertible in Z n . Consider the left multipli- cation map L ŒaŁ W Z n ! Z n defined by L ŒaŁ .ŒbŁ/ D ŒaŁŒbŁ D ŒabŁ . Since ŒaŁ is not invertible, Œ1Ł is not in the range of L ŒaŁ , so L ŒaŁ is not surjective. Conclude that L ŒaŁ is not injective, and use this to show that there exists ŒbŁ ¤ Œ0Ł such that ŒaŁŒbŁ
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: D 0 in Z n . 1.7.14. Suppose a is relatively prime to n . (a) Show that for all b 2 Z , the congruence ax b . mod n/ has a solution. (b) Can you nd an algorithm for solving congruences of this type? Hint: Consider Exercise 1.7.11 . (c) Solve the congruence 8x 12 . mod 125/ . 1.7.15. This exercise guides you to a proof of the Chinese remainder the-orem. (a) To prove the Chinese remainder theorem, show that it sufces to nd m and n such that C ma D C nb . (b) To nd m and n as in part (a), show that it sufces to nd s and t such that as C bt D . / . (c) Show that the existence of s and t as in part (b) follows from a and b being relatively prime. 1.7.16. Find and integer x such that x 3 . mod 4/ and x 5 . mod 9/ . 1.8. Polynomials Let K denote the set Q of rational numbers, the set R of real numbers, or the set C of complex numbers. ( K could actually be any eld ; elds are...
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online