PMATH 340 Number Theory, Assignment 4 Due: Tues July 21 Read Chapters 6 and 7 in the Lecture Notes, and work on the Exercises for Chapters 6 and 7 in the Practice Problems. Then solve each of the following problems. 1: (a) Factor 3 + 2 √ 3 i as a product of irreducible elements in the ring Z [√ 3 i ] . (b) Show that 4 + √ 5 i is irreducible but not prime in the ring Z [√ 5 i ] . (c) Show that the ring Z [√ 2 ] is a unique factorization domain. 2: (a) Without proof, list all of the irreducible elements z ∈ Z[√6 i] with ||z|| ≤ 10. (b) Without proof, list all of the elements z ∈ Z[√6 i] with ||z|| ≤ 10 which do not factor uniquely. (c) Let p be a prime in Z+. Show that p is reducible in Z[ √ 6 i] if and only if p = x2+6y2 for some x, y ∈ Z+. 3: (a) Express the (periodic) continued fraction [1, 3, 1, 1, 2 ] as a quadratic irrational. (b) Find the 4th convergent c4 = p4 q4 for the continued fraction representation of e2. (c) Express √ 43 as a continued fraction and find the smallest unit u > 1 in Z[ √ 43 ].
