Math 171: Abstract Algebra, Fall 2014- Assignment 11
1. In this problem, we will use rings to prove that there are infinitely many pairs of integers (a, b) such that a2 - 2b2 = 1. Consider the ring R = Z[√2]. For each element a + b√2 ∈ R, define N(a + b√2) = a2 - 2b2.
(a) Show that N is not a norm on R, but prove that for all x, y ∈ R, N(xy) = N(x)N(y).
(b) If N(x) = ±1, prove x is a unit in R. Moreover, find a unit u in R.
(c) Deduce that every element of the set {uk| k ∈ Z≥1} is a unit in R.
(d) Find infinitely many pairs of integers (a, b) for which a2 - 2b2 = 1.
2. Let R be a PID and I be a prime ideal of R. Prove R/I is a PID.
3. Read the subsection "Factorization in the Gaussian Integers" on pages 289-292 in Dummit & Foote. Using this, determine all representations of the integer 2130797 = 172 · 73 · 101 as the sum of two squares of integers.
4. (a) If x ∈ Z[i] and N(x) is a prime number in Z, prove x is irreducible.
(b) Factor the element 45 in Z[i] into a product of irreducibles.