Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 6916. Convert the following complex numbers to the form a + bi.(a) 2 cis(π/6)(c) 3 cis(π)(b) 5 cis(9π/4)(d) cis(7π/4)/217. Change the following complex numbers to polar representation.(a) 1 − i(c) 2 + 2i(e) −3i18. Calculate each of the following expressions.(b) −5(d) √ 3 + i(f) 2i + 2 √ 3(a) (1 + i) −1(c) ( √ (b) (1 − i) 63 + i) 5(d) (−i) 10(e) ((1 − i)/2) 4(f) (− √ 2 − √ 2 i) 12(g) (−2 + 2i) −519. Prove each of the following statements.(a) |z| = |z|(c) z −1 = z/|z| 2(e) |z − w| ≥ ||z| − |w||(b) zz = |z| 2(d) |z + w| ≤ |z| + |w|(f) |zw| = |z||w|20. List and graph the 6th roots of unity. What are the generators of this group?What are the primitive 6th roots of unity?21. List and graph the 5th roots of unity. What are the generators of this group?What are the primitive 5th roots of unity?22. Calculate each of the following.(a) 292 3171 (mod 582)(c) 2071 9521 (mod 4724)23. Let a, b ∈ G. Prove the following statements.(b) 2557 341 (mod 5681)(d) 971 321 (mod 765)(a) The order of a is the same as the order of a −1 .(b) For all g ∈ G, |a| = |g −1 ag|.(c) The order of ab is the same as the order of ba.24. Let p and q be distinct primes. How many generators does Z pq have?25. Let p be prime and r be a positive integer. How many generators does Z p rhave?26. Prove that Z p has no nontrivial subgroups if p is prime.