The way I define the Eisenstein integers is by letting and then defining .
Originally Posted by qpal1234
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 .
b) Show that the sum, difference, and product of two Eisenstein integers is also an Eisenstein integers
Using the hint show using the above fact it follows .
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)
If then . Now just show by multipling out that , again use the fact .
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
Say then you can write . Multiply out, . Thus, . Now determine if this is possible for integers. Do the same with other equation.
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.
Hint: prove that an Eisenstein integer is a unit if and only if its norm is 1.
f) An Eisenstein integer ε is a unit if ε divides 1. Find all the Eisenstein integers that are units
Hint: if is a prime then where is prime.
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.
Let then divide where . Now let be such that . Then form . Prove that or .
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
We have just proven above that is an Euclidean domain, so it is a unique factorization domain.
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