Proof that The Gaussian Integers With + is a group

We need to show that The Gaussian Integers With + is associative, there exists an identity element, and there exists inverses.

Associativity: Due to the nature of the definition of a complex number, we know that the complex numbers are associative.

So we know that for all a, b, c, d in Z[i], we have:

(2)
\begin{equation} (a+bi)+(c+di)=(a+c)+(b+d)i \end{equation}

Therefore the Gaussian Integers with + are associative.

Identity Element: If we let a = 0 and b = 0, we see that

(3)
\begin{equation} (a+bi)=(0+0i)=(0+0)=0 \end{equation}

So we have,

(4)
\begin{equation} (a+bi)+0= (a+bi) \end{equation}

Therefore, 0 is the identity element of The Gaussian Integers with +.

Inverse Element: Let us assume that there exists Q in Z[i] and Q^{-1} in Z[i] such that Q=x+iy and Q^{-1}=a+bi for all integers a, b, x, y

Then we have

(5)
\begin{equation} Q+Q^{-1}=(x+iy)+(a+bi)=0 \end{equation}

Therefore,

(6)
\begin{equation} (x+a)+i(y+b)=0 \end{equation}

Hence,

(7)
\begin{align} x+a=0 $\ i(y+b)=0 \end{align}

So

(8)
\begin{align} x=-a $\ y=-b \end{align}

Therefore, the inverse element exists for (a+bi) and it is (-a-bi).

So the Gaussian Integers with + is a group.