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# System prepended metadata

title: MTH 350 Homework 8
tags: [' homework', mth350]

---

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tags: mth350, homework
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# MTH 350 Homework 8

Instructions for this Homework set are on Blackboard where this is posted. The due date is on the class calendar, which is linked on Blackboard. 



## Problem 1

:::warning
This problem is an **independent** problem. As described in [the Syllabus](https://docs.google.com/document/d/1Rest_DodWnDy7Y8EVhp2lEOVnWcTEjLAGfWV2khuQQY/edit?usp=sharing) (top of page 14) this means that **on this problem, the only help you can get is from the professor.** 
:::

The set of **Gaussian integers**, denoted $\mathbb{Z}[i]$, is the following subset of the complex numbers $\mathbb{C}$: 

$$\mathbb{Z}[i] = \left\{ a + bi \in \mathbb{C} \, : \, a,b \in \mathbb{Z} \right\}$$

In other words, $\mathbb{Z}[i]$ consists of all complex numbers whose real and imaginary parts are integers. Therefore, for example $3 - 8i \in \mathbb{Z}[i]$ but $\frac{1}{2} + 2i \not \in \mathbb{Z}[i]$. 

This problem has multiple parts: 

1. Prove that $\mathbb{Z}[i]$ is a subring of $\mathbb{C}$. 
2. **Prove or disprove**: $\mathbb{Z}[i]$ is a field. 
3. Give three examples of units in $\mathbb{Z}[i]$, at least one of which has a nonzero imaginary part. Prove that your examples work (i.e. that they really are units). If this isn't possible, explain why. 
 


**Notes and hints:**

- Because of part 1 here, we know $\mathbb{Z}[i]$ is a ring. In fact, it's easy to see that $\mathbb{Z}[i]$ is not just a ring but a commutative ring with identity. (The identity in $\mathbb{C}$ is $1 + 0i$, and this is in $\mathbb{Z}[i]$ because both the real and imaginary parts are integers; we know multiplication is commutative because it's commutative in $\mathbb{C}$, and commutativity is inherited.) How much more would it take to prove $\mathbb{Z}[i]$ is a field? 
- For parts 2 and 3, the proof of [Problem 2.1](/oRoUklXnR76NWhKvVFy11A), which was done in class, might be useful. In particular, there is a formula in that proof that shows how to form the multiplicative inverse of a complex number. Practice using it on some elements of $\mathbb{Z}[i]$ before attempting any proof. 
- Remember that if you are asked to give an example of something, or if you are demonstrating a counterexample that disproves a statement, the example must be **specific** and **concrete**. For example if you think $\mathbb{Z}[i]$ is *not* a field, you'll need to show that one of the field axioms isn't satisfied, and this requires demonstration of a *specific number* or pair of numbers that leaves no doubt that the axiom fails. General arguments are for proofs; specific examples are for examples. 

## Problem 2
:::success
This problem is a **collaborative** problem. This means that **you can work on it in a small group, as long as you stay within the bounds of academic honesty** laid out in [the Syllabus](https://docs.google.com/document/d/1Rest_DodWnDy7Y8EVhp2lEOVnWcTEjLAGfWV2khuQQY/edit?usp=sharing) (starting on page 13). 
:::

Work **exactly one** of the following: 

1. Prove or disprove: Let $\mathbb{Z}[i]$ be the ring of Gaussian integers you saw in Problem 1. Are there any zero divisors in $\mathbb{Z}[i]$? If you think "yes", then give an example and prove that your example is correct. If you think "no", prove it (i.e. that $\mathbb{Z}[i]$ has no zero divisors). 
2. Give a complete and correct solution for [Problem 2.10](/oRoUklXnR76NWhKvVFy11A).


## Submission instructions

Turn in a single PDF with: 

- Your solution for all three parts of Problem 1
- Your solution for Problem 2

Remember to type up your work using $\LaTeX$, and to put a page break in between the solution for Problem 1 and the solution for Problem 2. 

Then upload your PDF to the "Homework 8" assignment area, and remember to hit **Submit**.

:::warning
**Reminder**: You will be given feedback on your work **only if both solutions are complete, good-faith efforts at correct work.** Work that is partial (for example doing only the second two parts of Problem 1) or insubstantial will result in the entire submission being returned to you with no comment, and you'll need to spend a revision to resubmit before getting any feedback on it.
:::

 

**Handy $\LaTeX$ commands:** 

The code for the Gaussian integer symbol is: 

```latex
$\mathbb{Z}[i]$
```

If I were typing this up, I'd make a macro for it in the preamble of my document. 