Math 361, Spring 2021, Assignment 12

Read:

1. Section 23.

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

1. Irreducible element (of a domain).
2. Unique factorization domain.

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

1. Factor Theorem (you proved this on a previous homework assignment).
2. Theorem concerning polynomials of degree two or three that have no roots.
3. Eisenstein's Criterion (this is Theorem 23.15 in the text).

Carefully describe the following procedures:

1. Sieve of Eratosthenes (to find irreducible positive integers up to a predetermined value).
2. Sieve of Eratosthenes (to find irreducible polynomials in $F[x]$, where $F$ is a finite field, up to a predetermined degree).

Solve the following problems:

1. Section 23, problems 9, 10, 11, 12, 13, 14, 19, 21, 29, and 30.
2. Recall the norm function $N:\mathbb{Z}[\sqrt{-5}]\rightarrow\mathbb{Z}$ defined by the formula $N(a+bi\sqrt{5})=a^2+5b^2$. Show that every element of norm $1$ is a unit of $\mathbb{Z}[\sqrt{-5}]$.
3. Compute $N(1+i\sqrt{5})$ and $N(1-i\sqrt{5})$.
4. Show that $\mathbb{Z}[\sqrt{-5}]$ has no elements of norm $2$ and no elements of norm $3$.
5. Prove that $1+i\sqrt{5}$ is an irreducible element of $\mathbb{Z}[\sqrt{-5}]$. (Hint: for any elements $z_1,z_2\in\mathbb{Z}[\sqrt{-5}]$, we have $N(z_1z_2)=N(z_1)N(z_2)$.) Similarly, prove the $1-i\sqrt{5}$ is an irreducible element.
6. Use similar techniques to prove that $2$ and $3$ are both irreducible in $\mathbb{Z}[\sqrt{-5}]$.
7. Prove that $\mathbb{Z}[\sqrt{-5}]$ is not a unique factorization domain. (Hint: in the lectures we gave two essentially different factorizations of $6$ in this ring; now you know that these really were "complete" factorizations, i.e. factorizations into irreducible elements.)

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