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College Algebra Exam Review 34

# College Algebra Exam Review 34 - D Œ0Ł in Z n 1.7.14...

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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
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Unformatted text preview: D Œ0Ł in Z n . 1.7.14. Suppose a is relatively prime to n . (a) Show that for all b 2 Z , the congruence ax ² b . mod n/ has a solution. (b) Can you ﬁnd an algorithm for solving congruences of this type? Hint: Consider Exercise 1.7.11 . (c) Solve the congruence 8x ² 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 sufﬁces to ﬁnd m and n such that ˛ C ma D ˇ C nb . (b) To ﬁnd m and n as in part (a), show that it sufﬁces to ﬁnd s and t such that as C bt D .˛ ³ ˇ/ . (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 ² 3 . mod 4/ and x ² 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 ﬁeld ; ﬁelds are...
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