Math 361, Spring 2018, Assignment 1

Carefully define the following terms, then give one example and one non-example of each:

1. Field.
2. Zero-divisor.
3. Integral domain.
4. Divisibility relation.
5. Associate relation.
6. Associate class.
7. Equivalent factorizations.
8. Trivial factorization.
9. Irreducible element.
10. Unique factorization domain.
11. Characteristic of a ring (see pp. 181-182; we did not discuss this in class).

Carefully state the following theorems (you do not need to prove them):

1. Theorem relating integral domains to fields.
2. Zero-product property (of an integral domain).
3. Cancellation law (in an integral domain).
4. Theorem concerning the basic properties of the divisibility relation.
5. Theorem concerning the basic properties of the associate relation.
6. Theorem describing the associate class of an element (in terms of units).
7. Theorem relating divisibility by an element to divisibility by an associate.
8. Theorem relating divisibility of an element to divisibility of an associate.

Solve the following problems:

1. Section 19, problems 1, 2, 3, 4, 5, 8, and 9. (Warning: $\mathbb{Z}_6$ and $\mathbb{Z}_{12}$ are not integral domains, so it is not safe to solve problems 1, 3, and 4 by methods of high school algebra.)
2. Suppose that $n$ is a composite number, say $n=ab$ for integers $a,b>1$. Prove that $\mathbb{Z}_n$ is not an integral domain.
3. Suppose that $n$ is a composite number, say $n=ab$ for integers $a,b>1$. Prove that $\mathbb{Z}_n$ is not a field.
4. Find four distinct units in the ring of Gaussian integers $\mathbb{Z}[i]$. (Later in the semester we will show that these are the only units in $\mathbb{Z}[i]$. For the problem below you may take this fact for granted.)
5. Working in the ring of Gaussian integers $\mathbb{Z}[i]$, compute the associate class $[2]$. Draw the class in the complex plane.
6. Working in the ring of Gaussian integers $\mathbb{Z}[i]$, compute the associate class $[2+3i]$. Draw the class in the complex plane.
7. Working in $\mathbb{Z}[i]$, show that the integer $5$ is not irreducible (i.e. find a non-trivial factorization of $5$).

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