Math 360, Fall 2013, Assignment 14

What would life be without arithmetic, but a scene of horrors?

- Sydney Smith

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

1. Degree (of a formal polynomial).
2. Divisibility relation (in a domain $D$).
3. Associate relation (in a domain $D$).
4. Irreducible element.
5. Unique factorization domain.

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

1. Formula for the degree of a product in $D[x]$ (where $D$ is a domain).
2. Characterization of the units of $D[x]$.
3. Fundamental Theorem of Arithmetic.
4. Theorem on unique factorization in $F[x]$ (where $F$ is a field).

Solve the following problems:

1. Let $R$ be a commutative unital ring, and let $f,g\in R[x]$. Show that $\mathrm{deg}(f+g)\leq\mathrm{max}(\mathrm{deg}(f),\mathrm{deg}(g))$, with equality holding whenever $\mathrm{deg}(f)\neq\mathrm{\deg}(g)$. Then give an example to show that strict inequality may occur when $f$ and $g$ have equal degrees.

1. Working in $\mathbb{Q}[x]$, show that the polynomial $f = x^2-x-2$ admits the factorization $f = (x+1)(x-2)$ and also admits the factorization $f = (2x+2)((1/2)x-1)$. Why doesn't this violate the theorem on unique factorization of polynomials?

1. Working in $\mathbb{Q}[x]$, show that the polynomial $f = x^3 + 6x^2 + 11x + 6$ admits the factorization $f = (x^2+3x+2)(x+3)$ and also admits the factorization $f = (x^2+5x+6)(x+1)$. Why doesn't this violate the theorem on unique factorization of polynomials?

1. Let $D=\{a+b\sqrt{5}i|a,b\in\mathbb{Z}\}.$ Show that $D$ is a unital subring of $\mathbb{C}$, and hence a domain. Then find two apparently different non-trivial factorizations of the element $6$ in $D$. (Hint: one factorization is obvious. To find another, note that $6$ can be written as the difference of two squares in $D$.) Then show that neither element in your second factorization is an associate of either element in your first. (In the spring we will show that all the elements in these two factorizations are in fact irreducible in $D$; it follows that $D$ does not have unique factorization.)

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

1.) Degree (of a formal polynomial).

If $p\in R\left[x\right]$ and $p\neq 0$, we write:

$$p= p_0 + p_1x+p_2x^2+...+p_nx^n$$

with $p_n\neq 0$ and only finitely many $p_i \neq 0$

The integer n is called "the degree of p", denoted $deg(p)$.

Example: $deg(x^4+2x^3+3x^2-6)= 4$

Non-example: $deg(4x^3+2x^2+18)\neq 4$