The Gaussian integers, denoted by $\mathbb{Z}[i]$, are the set of complex numbers of the form $a + bi$, where $a$ and $b$ are integers, and $i$ is the imaginary unit with the property $i^2 = -1$.

A unit in a ring is an element that has a multiplicative inverse within the ring. In other words, an element $u$ in $\mathbb{Z}[i]$ is a unit if there exists another element $v$ in $\mathbb{Z}[i]$ such that $u \times v = 1$.

To show that $\pm 1$ and $\pm i$ are the only units in $\mathbb{Z}[i]$, follow these steps:

1. Norm of a Gaussian Integer:

The norm of a Gaussian integer $z = a + bi$ is defined as: $N(z) = z \overline{z} = (a + bi)(a - bi) = a^2 + b^2$ where $\overline{z}$ is the complex conjugate of $z$. Notice that the norm $N(z)$ is always a non-negative integer.

2. Norm of a Unit:

For an element $u$ in $\mathbb{Z}[i]$ to be a unit, there must exist an element $v = c + di$ in $\mathbb{Z}[i]$ such that: $u \times v = 1$ Taking the norm on both sides: $N(u \times v) = N(1) = 1$ Since the norm is multiplicative: $N(u) \times N(v) = 1$ Given that $N(u)$ and $N(v)$ are non-negative integers, the only way their product can be 1 is if $N(u) = 1$ and $N(v) = 1$.

3. Norm Equals 1:

So, we need to find all Gaussian integers $u = a + bi$ such that: $N(u) = a^2 + b^2 = 1$ The equation $a^2 + b^2 = 1$ must be satisfied by integer values of $a$ and $b$. The possible pairs $(a, b)$ that satisfy this are:

• $a = 1, b = 0$, giving $u = 1$
• $a = -1, b = 0$, giving $u = -1$
• $a = 0, b = 1$, giving $u = i$
• $a = 0, b = -1$, giving $u = -i$

4. Conclusion:

Thus, the only Gaussian integers with a norm of 1 are $\pm 1$ and $\pm i$. These are the only elements in $\mathbb{Z}[i]$ that have multiplicative inverses within the ring. Therefore, $\pm 1$ and $\pm i$ are the only units in the ring of Gaussian integers $\mathbb{Z}[i]$.

Would you like more details or have any questions?

Here are five related questions to expand on this topic:

1. How do units in $\mathbb{Z}[i]$ relate to rotations in the complex plane?
2. What is the significance of the norm in other number rings, such as $\mathbb{Z}[\sqrt{-2}]$?
3. Can the Gaussian integers form a field, and if not, why?
4. How can we find the units in other quadratic integer rings, such as $\mathbb{Z}[\sqrt{2}]$?
5. What role do units play in the factorization of elements in $\mathbb{Z}[i]$?

Tip: When working with Gaussian integers, visualizing them in the complex plane as points can help in understanding their properties, especially the concept of norm and its role in identifying units.