Gaussian integers: GCD and Unique factorization

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Originally Posted by qpal1234

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]

The way I define the Eisenstein integers is by letting $\omega = e^{2\pi i/3}$ and then defining $Z[\omega ] = \{ a + b\omega | a,b\in \mathbb{Z} \}$ .

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b) Show that the sum, difference, and product of two Eisenstein integers is also an Eisenstein integers

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 $\omega^2 = -1 - \omega$ .

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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)

Using the hint show $\bar \omega = \omega^2$ using the above fact it follows $\omega^2 = -1-\omega \in Z[\omega]$ .

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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

If $\alpha = a + b\omega$ then $\bar \alpha = a + b\bar \omega = (a-b) - b\omega$ . Now just show by multipling out that $(a+b\omega)((a-b)-b\omega) = a^2 - ab + b^2$ , again use the fact $\omega^2 = -1 - \omega$ .

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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.

Say $1+2\omega | 1+5\omega$ then you can write $(1+2\omega) (a+b\omega) = 1 + 5\omega$ . Multiply out, $(a-2b)+(2a-b)\omega = 1+5\omega$ . Thus, $a-2b = 1 \mbox{ and }2a - b = 5$ . Now determine if this is possible for integers. Do the same with other equation.

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f) An Eisenstein integer ε is a unit if ε divides 1. Find all the Eisenstein integers that are units

Hint: prove that an Eisenstein integer is a unit if and only if its norm is 1.

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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.

Hint: if $\pi$ is a prime then $N(\pi) = p \mbox{ or }p^2$ where $p\in \mathbb{Z}^+$ is prime.

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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

Let $\beta \not = 0$ then divide $\tfrac{\alpha}{\beta} = p+q\omega$ where $p,q\in \mathbb{Q}$ . Now let $n,m\in \mathbb{Z}$ be such that $|n-p|,|m-q|\leq \frac{1}{2}$ . Then form $r = (n+m\omega)\beta - \alpha$ . Prove that $r=0$ or $N(r) < N(\beta)$ .

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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

We have just proven above that $Z[\omega]$ is an Euclidean domain, so it is a unique factorization domain.