13. 14. 15. 16. 17. Compute edz, where the circle is parametrized by t 2et, 0t 2. |z|=2 Leto and u denote the Euler totient function and Mbius function respectively. Q(n) (k) Show that n = (d). Hence show that - k d/n = n k/n Define conjugacy class in a finite group G and show that the cardinality of any conjugacy class divides the order of G. Use this to show that if p is a prime and G is a group of order p", then the centre of G contains elements other than the identity. Let I, I', J be ideals in a commutative ring A. If I, J are comaximal, i.e., I + J = A and I, J are comaximal, i.e., I' + J = A, then show that I I' and J are also comaximal. Let LK be a finite field extension of prime degree p. Show that L = K[a] for any a L\K.