Mathematical Modelling for Computer NetworksPart II Autumn 2013

Exercise 1: Due on 1st November 2013.

Attempt all the questions. Write your answers to the questions briefly and clearly. Please bring a printout (or a handwritten copy) of your answers to the class. We discuss the related topics in detail in the exercise class. You may refer to the book V. Shoup. A Computational Introduction to Number Theory and Algebra (link in the course page). 1. Construct the addition and multiplication tables for the residue classes modulus n, denoted by Z/nZ, for n = 2, 3 and 4. Identify those values of n which correspond to fields. 2. Show that Z/5Z is a field but Z/6Z is not. Can you state when the general case Z/nZ gives a field? If possible, prove the result. 3. Show that for integers a and b, (a + b)p = ap + bp mod p for any prime p. 4. Show that 2(p−1) = 1 mod p for any prime p 5. Calculate the entropy of tossing a fair coin and entropy for throwing a 6-faced fair die. 6. Which of the following polynomials over F2 [x], the ring of polynomials over the field F2 , are irreducible and find their factorization when possible. (a) x2 + 1, (b) (b)x2 + x + 1, (c) x2 + x and (d) x3 + 1

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Exercise 1: Due on 1st November 2013.

Attempt all the questions. Write your answers to the questions briefly and clearly. Please bring a printout (or a handwritten copy) of your answers to the class. We discuss the related topics in detail in the exercise class. You may refer to the book V. Shoup. A Computational Introduction to Number Theory and Algebra (link in the course page). 1. Construct the addition and multiplication tables for the residue classes modulus n, denoted by Z/nZ, for n = 2, 3 and 4. Identify those values of n which correspond to fields. 2. Show that Z/5Z is a field but Z/6Z is not. Can you state when the general case Z/nZ gives a field? If possible, prove the result. 3. Show that for integers a and b, (a + b)p = ap + bp mod p for any prime p. 4. Show that 2(p−1) = 1 mod p for any prime p 5. Calculate the entropy of tossing a fair coin and entropy for throwing a 6-faced fair die. 6. Which of the following polynomials over F2 [x], the ring of polynomials over the field F2 , are irreducible and find their factorization when possible. (a) x2 + 1, (b) (b)x2 + x + 1, (c) x2 + x and (d) x3 + 1

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