Domains and Factorization

Let $D$ be an integral domain and $a,b\in D$. Prove that $a\mid b$ and $b\mid a$ if and only if $a$ and $b$ are associates.

Prove 1.11.

Let $d$ be a square-free integer that is not 1. Prove that $x\in\mathbb{Z}[\sqrt{d}]$ is irreducible if $N(x)=p$, where $p$ is a prime number.

Prove that multiplication over the field of fractions is a well-defined operation, and that it is commutative.

Prove or disprove the following statements.

1. $\mathrm{Frac}(\mathbb{Z}[x])\cong\mathbb{Q}[x]$.

2. $\mathrm{Frac}(k\mathbb{Z})\cong\mathbb{Q}$ for all non-zero $k\in\mathbb{Z}$.

Let $D$ be an integral domain, $a,b\in D$, and $b\neq 0$. Prove that $\left\langle ab\right\rangle\subset\left\langle b\right\rangle$ if and only if $a$ is not a unit.(Hint: consider contrapositives.)

Prove 1.38.

Prove 1.45.

Cite the appropriate theorems to explain the rest of the class inclusions in 1.1.

Prove that if a prime $p$ can be written as $a^{2}+b^{2}$ where $a$ and $b$ are integers, then $a+bi$ is a Gaussian prime.

Let $D$ be an integral domain and let $p\in D$ be a prime. Suppose the elements $a_{1},a_{2},a_{3},\dots,a_{n}\in D$ are such that $p\mid a_{1}a_{2}a_{3}\dots a_{n}$, where $n\geq 2$. Prove that $p\mid a_{i}$ for some $i\in\{1,2,\dots,n\}$.

Determine whether the following Gaussian integers are Gaussian primes, proving your claims.

1. 6

2. 7

3. 13

4. $1+2i$

5. $3+4i$

6. $5+i$

Let $D$ be an integral domain. Define a relation $R$ on $D$ such that $a\mathrel{R}b$ if and only if $a$ and $b$ are associates. Prove that $R$ is an equivalence relation.

Expand the polynomials $(3x+2)(x+4)$ and $(4x+1)(2x+3)$ in $\mathbb{Z}_{5}[x]$. Does this result contradict 1.32?

Prove 1.8.

Let $F$ be a field. Prove that $\mathrm{Frac}(F)\cong F$.

Prove that $\mathbb{Z}[\sqrt{-6}]$ is not a UFD.(Hint: show that some specific elements are irreducible, and then factor 6 in two ways.)

Let $D$ be a UFD and let $p\in D$. Prove that $p$ is irreducible if and only if $p$ is prime.

Let $D$ be a Euclidean domain with norm $N$. Prove the following statements.

1. If $a,b\in D$ are associates then $N(a)=N(b)$.

2. $u$ is a unit in $D$ if and only if $N(u)=N(1)$.

Suppose $D$ is an integral domain and $\phi:D\to\mathbb{N}\cup\{0\}$ is a non-constant function such that $\phi(xy)=\phi(x)\phi(y)$ for all $x,y\in D$. Prove that if $u\in D$ is a unit then $\phi(u)=1$.

Prove that, for any field $F$, there exists infinitely many irreducible polynomials in $F[x]$ that are not associates.

Prove that for every non-trivial ideal $I$ of the Gaussian integers, the quotient ring $\mathbb{Z}[i]/I$ is finite.

Consider the polynomial

in the ring $\mathbb{Z}[x]$.

1. Explain why the coefficient of the term of degree $k$ (where $2\leq k\leq 12$) in $P(x)$ gives exactly the number of ways that a pair of two dice could sum to $k$.

2. Factor $P(x)$ into a product of irreducible polynomials. Prove that the factorization, indeed, consists only of irreducible polynomials.

3. Define the polynomials

   $\displaystyle f(x)$ $\displaystyle=x^{a_{1}}+x^{a_{2}}+x^{a_{3}}+x^{a_{4}}+x^{a_{5}}+x^{a_{6}}\text{ and}$ $\displaystyle g(x)$ $\displaystyle=x^{b_{1}}+x^{b_{2}}+x^{b_{3}}+x^{b_{4}}+x^{b_{5}}+x^{b_{6}},$

   where each $a_{i}$ and $b_{i}$ are positive integers. Suppose w…