Math 361, Spring 2022, Assignment 13

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

1. Prime ideal.
2. Unique factorization domain.
3. Principal ideal domain.
4. Prime element.
5. Proper divisor chain.
6. Divisor chain condition.

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

1. Theorem relating prime ideals to integral domains.
2. Theorem relating irreducible elements to maximal ideals ("If $D$ is a principal ideal domain, then $\left\langle m\right\rangle$ is maximal if and only if $m$ is...").
3. Theorem relating prime elements to irreducible elements in general.
4. Theorem relating prime elements to irreducible elements in principal ideal domains.
5. Criteria for $D$ to have unique factorization.
6. Classification of ideals in $\mathbb{Z}$ ("$\mathbb{Z}$ is a...").
7. Theorem concerning divisor chains in $\mathbb{Z}$ ("$\mathbb{Z}$ has no...").
8. Theorem concerning unique factorization in $\mathbb{Z}$.
9. Classification of ideals in $F[x]$ ("For any field $F$, $F[x]$ is a...").
10. Theorem concerning divisor chains in $F[x]$ ("For any field $F$, $F[x]$ has no...").
11. Theorem concerning unique factorization in $F[x]$.

Solve the following problems:

1. The polynomial $f=x^3+x$ has an essentially unique factorization into primes of $\mathbb{R}[x]$. Find this factorization.
2. The polynomial $f=x^3+x$ has an essentially unique factorization into primes of $\mathbb{C}[x]$. Find this factorization.
3. (The domain $\mathbb{Z}[\sqrt{-5}]$) Recall that $\mathbb{Z}[\sqrt{-5}]=\{a+bi\,|\,a,b\in\mathbb{Z}\}$. Show that this set is a unital subring of $\mathbb{C}$, and hence an integral domain.
4. Define a function $N:\mathbb{Z}[\sqrt{-5}]\rightarrow\mathbb{Z}_{\geq0}$ by the formula $N(z)=\left\lvert z\right\rvert^2$. (Here the absolute value is taken in the sense of complex numbers, i.e. $\left\lvert a+bi\right\rvert=\sqrt{a^2+b^2}$.) Show that $N$ preserves multiplication, i.e. that $N(z_1z_2)=N(z_1)N(z_2)$.
5. Find all elements $a+bi\sqrt{5}\in\mathbb{Z}[\sqrt{-5}]$ with $N(a+bi\sqrt{5})=1$.
6. Show that an element of $\mathbb{Z}[\sqrt{-5}]$ is a unit if and only if it has norm one.
7. Show that in the ring $\mathbb{Z}[\sqrt{-5}]$, the factorization $a=bc$ is non-trivial if and only if $N(b)<N(a)$ and $N(c)<N(a)$.
8. Show that $\mathbb{Z}[\sqrt{-5}]$ has no elements of norm $2$. (Hint: $N(a+bi\sqrt{5})=a^2+5b^2$, and both $a$ and $b$ are integers.)
9. Show that $\mathbb{Z}[\sqrt{-5}]$ has no elements of norm $3$.
10. Calculate the norms of the elements $2, 3, 1+i\sqrt{5},$ and $1-i\sqrt{5}$.
11. Show that all four of the elements referenced in the previous problem are irreducible in $\mathbb{Z}[\sqrt{-5}]$.
12. Show that none of the elements referenced above is prime.
13. Show that $\mathbb{Z}[\sqrt{-5}]$ does not have unique factorization.
14. Show that $\mathbb{Z}[\sqrt{-5}]$ does satisfy the divisor chain condition. (Hint: think about norms in a proper divisor chain.)
15. Show that $\mathbb{Z}[\sqrt{-5}]$ must contain at least one non-principal ideal.
16. Consider the ideal $J=\left\langle 2,1+i\sqrt{5}\right\rangle=\{2(a+bi\sqrt{5})+(1+i\sqrt{5})(c+di\sqrt{5})\,|\,a,b,c,d\in\mathbb{Z}\}=\{(2a+c-5d)+(2b+c+d)i\sqrt{5}\,|\,a,b,c,d\in\mathbb{Z}\}$. Show that $2\in J$ and $1+i\sqrt{5}\in J$ but $1\not\in J$. (Hint: to show that $1\not\in J$, work with the last-given description of the elements of $J$. In order for the coefficient of $i\sqrt{5}$ to vanish, $c$ and $d$ must both be even or both odd. In either case, what is the parity of $2a+c-5d$?)
17. Show that the ideal $J$ is not principal. (Hint: if it were principal, say $J=\left\langle g\right\rangle$, then the generator $g$ would need to be a common divisor of $2$ and $1+i\sqrt{5}$. But these are irreducibles and are not associates of one another. So what are their common divisors?)
18. (Optional; a domain with an infinite divisor chain) This and all following exercises require some knowledge of complex analysis and are thus optional. In these exercises, if you choose to attempt them, you will construct an example of an infinite proper divisor chain. To begin with, let $R$ denote the set of functions from $\mathbb{C}$ to $\mathbb{C}$ which are complex-analytic at every point. Using the properties of complex derivatives, show that $R$ is a unital ring under pointwise addition and multiplication.
19. Show that a non-constant element of $R$ can vanish at only countably many points. (Hint: this is the hardest exercise of the whole series. You will need to use the identity theorem together with the fact that $\mathbb{C}$ is a second-countable topological space.)
20. Show that $R$ is an integral domain. (Hint: if $fg=0$ then either $f$ or $g$ must vanish at uncountably many points.)
21. Show that a unit of $R$ cannot have any zeros. (Hint: if $f$ has a zero of order $d$ at $z=z_0$, then $1/f$ has a pole of order $d$ at $z=z_0$.)
22. Define a function $f_n:\mathbb{C}\rightarrow\mathbb{C}$ by the formula $f(z)=\sin(z)/\prod_{k=1}^n\left(z-k\pi\right)$. Show that $f_n$ has only removable singularities and thus has a unique extension to an element of $R$ (which we shall also denote by $f_n$).
23. Describe the zeros of $f_n$.
24. Show that $f_n/f_{n+1}$ has only removable singularities, and thus $f_{n+1}\,|\,f_n$.
25. Show that $f_{n+1}\not\sim f_n$. (Hint: use the principle, which you proved above, that a unit of $R$ cannot have any zeros.)
26. Conclude that $(f_1,f_2,f_3,\dots)$ is an infinite proper divisor chain in $R$.

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