How can we show that ±1,±i are the only units in the ring of Gaussian integers, Z[i]?

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.

Explore why ±1 and ±i are the only units in the ring of Gaussian integers (Z[i]). Understand the concept of units in rings, norm of Gaussian integers, and their implications. Detailed solution and explanation provided.

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