Math 361, Spring 2020, Assignment 9

Read:

1. Section 27.

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

1. Trivial factorization (in a domain).
2. Irreducible element (of a domain).
3. Prime element (of a domain).
4. $\mathbb{Z}[\sqrt{-5}]$.
5. $N(z)$ (the norm of $z\in\mathbb{Z}[\sqrt{-5}]$).
6. Principal ideal domain (or PID for short).
7. Prime ideal.
8. Maximal ideal.

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

1. Factor theorem.
2. Practical method to factor a polynomial (given a root).
3. Criterion for irreducibility of polynomials of degree three or less.
4. Example of a high-degree polynomial which has no roots but is nevertheless educible.
5. Theorem relating irreducibility to primeness.
6. Theorem concerning the norm of a product (in $\mathbb{Z}[\sqrt{-5}]$).
7. Theorem concerning the units of $\mathbb{Z}[\sqrt{-5}]$.
8. Example of an irreducible element (of some domain) which is not prime.
9. Theorem concerning ideals of $F[x]$.
10. Eisenstein's Criterion (this is in section $23$ of the text).
11. Theorem relating prime ideals to integral domains.
12. Theorem relating primeness of the ideal $\left\langle a\right\rangle$ to primeness of the element $a$.
13. Theorem relating maximal ideals to fields.
14. Theorem relating maximality to primeness.
15. Example of a prime ideal which is not maximal.

Solve the following problems:

1. Section 23, problems 11, 12, 13, 14, 15, 16, 18, 19, 20, and 21.
2. Section 27, problems 1, 3, 5, 7, 18.
3. Suppose $F$ is a field. Find all ideals of $F$. (Hint: first show that $\{0\}$ is a maximal ideal.)
4. Suppose $R$ is a commutative, unital ring, not the zero ring, in which $\{0\}$ and $R$ itself are the only ideals. Prove that $R$ is a field.
5. (Failure of unique factorization) In elementary school, you learned that any given integer has many different factorizations (e.g. $12=3\times4=2\times6$) but only one factorization into irreducibles (e.g. $12=2\times2\times3$). Is this true in an arbitrary integral domain? (Hint: think about $\mathbb{Z}[\sqrt{-5}]$)
6. Actually factorization into irreducibles isn't really unique even in $\mathbb{Z}$. For example, $15=3\times5=5\times3=(-3)\times(-5)=(-5)\times(-3)$ exhibits four distinct factorizations of $15$ into irreducibles. Try to make a precise definition of "essentially unique" factorization into irreducibles so that factorization in $\mathbb{Z}$ is "essentially" unique.
7. Is factorization in $\mathbb{Z}[\sqrt{-5}]$ essentially unique, according to your definition above?

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