Math 361, Spring 2022, Assignment 11

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

1. Maximal ideal.
2. Divisibility relation (in a domain $D$; i.e. $a|b$ if and only if...).
3. Associate relation (in a domain $D$; i.e. $a\sim b$ if and only if...).
4. Associate class (of an element of a domain $D$).

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

1. Theorem characterizing the ideals $I$ for which $R/I$ is a field.
2. Containment criterion for principal ideals (i.e. $\left\langle a\right\rangle\subseteq\left\langle b\right\rangle$ if and only if...).
3. Equality criterion for principal ideals (i.e. $\left\langle a\right\rangle=\left\langle b\right\rangle$ if and only if...).
4. Properties of the associate relation (i.e. $\sim $ is an...).
5. Characterization of the associate class $[a]_\sim$
6. List of units in $\mathbb{Z}$.

Solve the following problems:

1. Suppose $u\in F[x]$ is a unit. Prove the $\mathrm{deg}(u)=0$. (Hint: start with the equation $u\cdot u^{-1}=1$, and take degrees of both sides.)
2. Suppose $f\in F[x]$ has degree zero. Show that $f$ is a unit. (Hint: remember that $F$ is a field. What sort of polynomials have degree zero?)
3. Let $D$ be any integral domain. Prove that $a\in D$ is a unit if and only if $a\sim 1$.
4. Suppose $u\in D$ is a unit and $u\sim v$. Prove that $v$ is also a unit.
5. An element $a\in D$ is said to be irreducible if it is not zero, not a unit, and given any factorization $a=bc$, either $b$ is a unit or $c$ is a unit. Describe the irreducible elements of $\mathbb{Z}$.
6. Working in $F[x]$ where $F$ is some field, show that any polynomial of degree one is irreducible. (Hint: suppose $\deg(f)=1$ and $f=gh$. Taking the degree of both sides of this equation gives $1=\deg(g)+\deg(h)$. What are all the possible values for the ordered pair $(\deg(g),\deg(h))$?)
7. Working in $F[x]$, show that a polynomial $f$ of degree two is irreducible if and only if it has no roots in $F$. (Hint 1: you will need the Factor Theorem that you proved in Assignment 9. Hint 2: suppose you have a factorization $f=gh$ in which neither $g$ nor $h$ is a unit. What are the degrees of $g$ and $h$?)
8. Working in $F[x]$, show that a polynomial $f$ of degree three is irreducible if and only if it has no roots in $F$. (Hint: this is very similar to the previous exercise.)
9. Give an example of a field $F$ and a polynomial $f\in F[x]$ of degree four, which has no roots but is nevertheless reducible. (Hint: this is much easier than it looks. The most familiar examples are those with $F=\mathbb{R}$. You simply need to find a pair of degree-two polynomials with no roots, and multiply them.)
10. Does the example you produced in the last problem invalidate the reasoning you used in the previous two? If not, at exactly what point does the reasoning you used in the previous two exercises break down in the case of degree-four polynomials?
11. Working once more in a general integral domain $D$, prove that if $a$ is irreducible and $a\sim b$, then $b$ is also irreducible.

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Questions:

1. Associate class (of an element of a domain $D$).

1. Equality criterion for principal ideals (i.e. $\left\langle a\right\rangle=\left\langle b\right\rangle$ if and only if...).

1. Characterization of the associate class $[a]_\sim$

1. List of units in $\mathbb{Z}$.

Definitions and Theorems:

https://drive.google.com/file/d/17jnBWbZgvWYUD5UW7kqw6aGYCj-BebE-/view?usp=sharing

Problems:

https://drive.google.com/file/d/1cQz6Su4YeTzI2dH4OGu8FfFq3fGUIE8H/view?usp=sharing