Gaussian integers

The field $\mathbb{Q}(\alpha)$ introduced above may be thought of as being similar to $\mathbb{Q}$ in many respects. What is the analog (if any) of the ring of integers, $\mathbb{Z}\subset \mathbb{Q}$? This subsection addresses this question.

Let $i=\sqrt{-1}$. The Gaussian number field is the special quadratic number field ${\mathbb{Q}}(i)$. The Gaussian integers is the ring

$${\mathbb{Z}}[i]=\{a+ib\ \vert\ a,b\in {\mathbb{Z}}\} ={\mathbb{Z}}+i{\mathbb{Z}}.$$

These ``integers'' can be pictured as an lattice in the complex plane:

Geometrically, addition of Gaussian integers may be visualized using the parallelogram law: if you represent the integer $a+ib$ by the vector $\vec{v}$ from $(0,0)$ to $(a,b)$ and the integer $c+id$ by the vector $\vec{w}$ from $(0,0)$ to $(c,d)$ then the sum $a+b+i(c+d)$ is represented by the vector from $(0,0)$ to the other endpoint of the diagonal of the parallelogram spanned by $\vec{v}$ and $\vec{w}$. For example, to add $1+i$ and $-1+1$, we have the following sketch

Definition 2.1.8   An element $p=a+ib\not= 0$ is a prime (element of $\mathbb{Z}[i]$) if it has the following property: for all $r,s\in \mathbb{Z}[i]$, $p\vert rs$ then $p\vert r$ or $p\vert s$.

The following characterization of Gausssian primes is known.

Theorem 2.1.9   An element $p=a+ib\not= 0$ is a prime element of $\mathbb{Z}[i]$ if and only if either

(a) $ab=0$ and $\vert a+ib\vert$ is a prime integer which is $\equiv 3\, ({\rm mod}\, 4)$, or

(b) $ab\not= 0$ and $a^2+b^2$ is a prime integer which is $\equiv 1\, ({\rm mod}\, 4)$.

The proof of this result goes beyond the scope of this book. See [HW] for a proof.

There is an analog of the fundamental theorem of arithmetic for the ring of Gaussian integers. In other words, each Gaussian integer can be uniquely (up to order and factors of the form $\pm 1$ and $\pm i$) factored into a product of Gaussian primes.

Remark 2.1.10   If you ask for a general analog of the fundamental theorem of arithmetic for the ring of integers of one of the number fields ${\mathbb{Q}}(\sqrt{d})$ then less is known. It was conjectured by Gauss (and still unproven today) that there are infinitely squarefree $d>0$ for which ring of integers of ${\mathbb{Q}}(\sqrt{d})$ satisfies the unique factorization theorem. It was proven by H. Stark in 1967 [St] (and independently by A. Baker) that the only cases for which ring of integers of ${\mathbb{Q}}(\sqrt{d})$, $d<0$ squarefree, satisfies the unique factorization theorem is when $d=-1,-2,-3,-7,-11,-19,-43,-67,-163$. See [HW], page 217, for further details.