a) Show that the algebraic integers of the form r+s√(-3), where r and s are rational numbers, are the numbers of the form a+bw, where a and b are integers and where w=(-1+√(-3))/2. Numbers of this form are called Eisenstein integers. The set of Eisenstein integers is denoted by Z[w]
b) Show that the sum, difference, and product of two Eisenstein integers is also an Eisenstein integers[/quote]
The addition property is straightforward. The only property that might not be as obvious as the others ones is multiplication. Here you need the fact that [tex]\omega^2 = -1 - \omega
The way I define the Eisenstein integers is by letting and then defining .
c) Show that if α is an Eisenstein integers, then α', the complex conjugate of α, is also n Eisenstein integers (Hint: First show that w'=w^2)
d) If α is an Eisenstein integers, we define the norm of this integer by N(α) = a^2-ab+b^2 if α = a+bw, where a and b are integers. Show that N(α) = α*(α') where α is an Eisenstein integers
e) If α and β are Eisenstein integers, we say that α divides β if there exists an element y in Z(w) such that β = α*y. Determine whether 1+2w divides 1+5w and whether 3+w divides 9+8w.
f) An Eisenstein integer ε is a unit if ε divides 1. Find all the Eisenstein integers that are units
g) An Eisenstein prime π in Z(w) is an element divisible only by a unit or an associate of π. (An associate of an Eisenstein integer is the product of that integer and a unit.). Determine whether each of the following elements are Eisenstein primes: 1+2w, 3-2w, 5+4w, and -7-2w. [/FONT][/SIZE][/FONT][/FONT]
h) Show that if α and β ≠ 0 belong to Z(w), there are number y and p such that α = βy + p and N(p)<N(β). That is establish a version of the division algorithm for the Eisenstein integers [/FONT][/SIZE][/FONT][/FONT][/quote]
i) Using part (h), show that Eisenstein integers can be uniquely written as the product of Eisenstein primes, with the appropriate considerations about associated primes taken into account[/FONT][/SIZE][/FONT][/FONT][/SIZE][/FONT]