Section A
Q1.
(a) Let F be a finite field of characteristic p, where p is a prime. Then show that there is an injective homomorphism from \mathbbZp (group of integers modulo p) to F. Also show that number of elements in F is pn, for some positive integer n.
(b) Let \mathbbR denote the set of real numbers and \mathbbQ denote the set of rational numbers. If x ∈ \mathbbR, x > 0 and y ∈ \mathbbR, then show that there exists a positive integer n such that nx > y. Use it to show that if x < y, then there exists p ∈ \mathbbQ such that x < p < y.
(c) Suppose f: [a, b] → \mathbbR is a continuous function. Then show that f is Riemann integrable on [a, b].
(d) Prove that the linear programming problem Maximize z = 3x1 + 2x2 subject to the constraints : 2x1 + x2 ≤ 2 3x1 + 4x2 ≥ 12 x1, x2 ≥ 0 does not admit an optimum basic feasible solution.
(e) Compute the integral ∮ (1 + 2z + z2)/((z-1)2(z + 2))dz where C is |z| = 3.
