How do I find $a,b\in\mathbb{Z}$ s.t. $\{ac-bd+i(ad+bc)\mid c, d\in\mathbb{Z}\}$ have real and imaginary parts both even or both odd?

Question

I'm trying to find some numbers, $a,b\in\mathbb{Z}$ s.t. the following equation is satisfied. \begin{equation} \{a c-b d+i(a d+b c) \mid c, d \in \mathbb{Z}\}=\{k+i l \mid k, l \text { even or odd }\} \end{equation}

So I need to find a number $a$ and $b$ s.t. the first set consist of complex numbers, where both the real part and imaginary part is even or the real and imaginary part is odd. Is it possible to find such numbers and how do I approach this?

Sorry about the sign variation on my prior post. But I didn't realise that the sign in front of bd was important.

Answer 1

Conceptually it boils down to: $\,\ \alpha\beta\,$ is even $\iff \alpha\,$ or $\,\beta\,$ is even, $ $ for $\,\alpha,\beta\in\Bbb Z[i],\,$ once we generalize the notion of "even" appropriately for Gaussian integers.

Hint for $\,p := (2,i\!-\!1)$ we have $\Bbb Z[i]/p \cong \Bbb Z/2\,$ so $\,p\,$ is prime & induces a parity structure on $\,\Bbb Z[i]\,$ via $\,\alpha := a+bi\,$ is $\rm\color{#c00}{even}$ $\iff p\mid\alpha \iff 2\mid a+b\iff a,b\,$ are $\rm\color{#c00}{equal\ parity}$. Hence

$$\begin{align} &\alpha\beta\,\ \text{is even},\ {\rm for}\,\ \beta = c+di\\[.2em] \iff\ &p\mid \alpha\beta\\[.2em] \iff\ & p\mid \alpha\,\ {\rm or}\ \,p\mid \beta,\,\ \text{by $\,p\,$ prime}\\[.2em] \iff\ &\alpha\,\ \text{is even or}\,\ \beta\,\ \text{is even}\\[.2em] \iff \ & a\equiv b\,\ {\rm or}\,\ c\equiv d\!\!\pmod{\!2}\end{align}\qquad\qquad$$

Answer 2

(1) Show that $I=\{k+li: 2|k-l\}$ is an ideal of $\mathbb Z[i]$. To show $I$ absorbs, the following characterization of $I$ is convenient: $$I=\{k+li: 2|k^2+l^2\}=\{z: |z|^2 \text{ is even } \}$$

(2) As $\mathbb Z[i]$ is a PID, $I$ does have a generator $a+bi$.

(3) $1+i\in I$, and $(1+i)\mathbb Z[i]$ is a maximal ideal of $\mathbb Z[i]$ (by e.g. $\mathbb Z[i]/(1+i)\cong \mathbb Z/2\mathbb Z$ is a field), hence $(1+i)\mathbb Z[i]=I$.

(4) Since $\{\pm 1, \pm i\}$ are all the units of $\mathbb Z[i]$, $a+bi$ can possibly be $\pm (1\pm i)$.