Math 361, Spring 2020, Assignment 10

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

1. Unique factorization domain (or UFD for short).
2. Divisor chain.
3. Monic polynomial.

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

1. Example of a domain which does not have unique factorization.
2. Theorem relating irreducible elements to maximal ideals in PIDs.
3. Theorem relating irreducibility to primeness in PIDs.
4. Theorem concerning divisor chains in $\mathbb{Z}$ and in $F[x]$.
5. Theorem concerning irreducible factors in domains without divisor chains.
6. Theorem concerning factorization into irreducibles in domains without divisor chains.
7. Theorem on unique factorization (relating unique factorization to divisor chains and primeness).
8. Fundamental Theorem of Arithmetic.
9. Theorem regarding unique factorization of polynomials.
10. Theorem concerning the number of elements in the ring $\mathbb{Z}_p[x]/\left\langle m\right\rangle$, where $\deg{m}=n$.
11. Theorem concerning monic associates of a given polynomial.

Describe the following algorithms:

1. Sieve of Eratosthenes (to generate lists of primes, either in $\mathbb{Z}$ or in $\mathbb{Z}_p[x]$).
2. Factorization algorithm in $\mathbb{Z}_p[x]$.
3. Algorithm to generate and implement many finite fields.

Solve the following problems:

1. Working in $\mathbb{Z}_3[x]$, factor the polynomial $x^3-x$ into irreducibles. (Hint: you do not need the Sieve.)
2. Working in $\mathbb{Z}_3[x]$, factor the polynomial $x^9-x$ into irreducibles. (Hint: you do not need the Sieve until quite late in the game. If you find yourself running the Sieve beyond degree $2$ then you are missing an opportunity. In fact, if you are very clever you do not need the Sieve at all.)
3. Describe at least three distinct fields, each with $9$ elements. (Warning: $\mathbb{Z}_3[x]/\left\langle x^2+1\right\rangle$ and $\mathbb{Z}_3[x]/\left\langle 2x^2+2\right\rangle$ are the same object: since the two polynomials are associates, they generate the same ideal.)
4. Use the Sieve to find all monic irreducibles in $\mathbb{Z}_2[x]$ up to degree $3$. Did you really need the Sieve to do that?
5. Describe at least two distinct fields, each with $8$ elements.
6. Describe a field with $16$ elements.
7. Describe a field with $343$ elements.

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