Question

HW 20. Due November 1. In this assignment, we will see an example of an integral domain that has elements that can be factored as a product of irreducible elements, but that factorization is not unique. Let R denote the set of all complex numbers a + b √ 5i, where a, b ∈ Z. Let N be the norm on R defined by N(a + b √ 5i) = a 2 + 5b 2 . As before N(z1z2) = N(z1)N(z2), for all z1, z2 ∈ R. (In fact, this holds for all complex numbers if, for z = c + di ∈ C we define N(z) = c 2 + d 2 .) (i) Show that R is an integral domain. (ii) Show that the only units in R are ±1. (iii) Use the norm to prove that 2, 3, 1 + √ 5i, 1 − √ 5i are irreducible elements in R. (iv) Conclude that 6 = 2 · 3 = (1 + √ 5i) · (1 − √ 5i) are two distinct factorizations of 4 into a product of irreducible elements.

Answer 1