Math 361, Spring 2019, Assignment 6

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

1. Divisibility relation (on an integral domain $D$).
2. Associate relation.
3. Irreducible element.
4. Unique factorization domain (a.k.a. UFD; for a non-example see problem 14 below).
5. Prime element.

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

1. Theorem concerning reflexivity and transitivity of the divisibility relation.
2. Theorem concerning reflexivity, symmetry, and transitivity of the associate relation.
3. Theorem characterizing the associate class of an element.
4. Theorem relating primeness to irreducibility.

Solve the following problems:

1. Let $F$ be any field, and suppose $f\in F[x]$ has degree one. Show that $f$ is irreducible. (Hint: suppose $f$ factors as $f=gh$, and think about the possible degrees of $g$ and $h$.)
2. Give an example of a field $F$ and an irreducible polynomial $f\in F[x]$ of degree greater than one.
3. Recall that $\mathbb{Z}[i]$ denotes the ring of Gaussian integers $\{a+bi\,|\,a,b\in\mathbb{Z}\}$. The norm of a Gaussian integer is defined as $N(a+bi)=a^2+b^2$. Observe that the norm is always a non-negative integer. Show that for two Gaussian integers $z_1$ and $z_2$, we have $N(z_1z_2)=N(z_1)N(z_2)$.
4. Suppose $u$ is a unit of $\mathbb{Z}[i]$. Show that $N(u)=1$. (Hint: take norms of both sides of the equation $uu^{-1}=1$.)
5. Find all units of $\mathbb{Z}[i]$, and compute their inverses.
6. List the elements of the associate class $[3+4i]$.
7. Show that although $5$ is irreducible when regarded as an element of $\mathbb{Z}$, it is not irreducible when regarded as a Gaussian integer. (Hint: compute the product $(1+2i)(1-2i)$, then observe that neither factor is a unit.)
8. Show that $1+2i$ is irreducible in $\mathbb{Z}[i]$. (Hint: suppose it factors as $1+2i=z_1z_2$ and think carefully about the possible norms of $z_1$ and $z_2$.) Do the same for $1-2i$.
9. Now consider the ring $\mathbb{Z}[\sqrt{-5}]=\{a+bi\sqrt{5}\,|\,a,b\in\mathbb{Z}\}$. Define a norm on this ring by the formula $N(a+bi\sqrt{5})=a^2+5b^2$. Show that $N(z_1z_2)=N(z_1)N(z_2)$.
10. Find all units of $\mathbb{Z}[\sqrt{-5}]$.
11. Show that $\mathbb{Z}[\sqrt{-5}]$ has no elements of norm two, and no elements of norm three.
12. Show that both $2$ and $3$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$, and compute their associate classes.
13. Show that both $1+i\sqrt{5}$ and $1-i\sqrt{5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$, and compute their associate classes.
14. Show that $\mathbb{Z}[\sqrt{-5}]$ does not have unique factorization.
15. Show that although $2$ is irreducible in $\mathbb{Z}[\sqrt{-5}]$, it is not prime there. Do the same for $3$, $1+i\sqrt{5}$, and $1-i\sqrt{5}$.

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