Abstract Algebra: Theory and Applications

Let $z=a+b\sqrt{3}i$ be in $\mathbb{Z}[\sqrt{3}i]\text{.}$ If $a^2+3b^2=1\text{,}$ show that $z$ must be a unit. Show that the only units of $\mathbb{Z}[\sqrt{3}i]$ are $1$ and $-1\text{.}$

Prove or disprove: Any subring of a field $F$ containing $1$ is an integral domain.

Prove or disprove: If $D$ is an integral domain, then every prime element in $D$ is also irreducible in $D\text{.}$

Let $F$ be a field of characteristic zero. Prove that $F$ contains a subfield isomorphic to $\mathbb{Q}\text{.}$

Let $p$ be prime and denote the field of fractions of ${\mathbb{Z}}_p[x]$ by ${\mathbb{Z}}_p(x)\text{.}$ Prove that ${\mathbb{Z}}_p(x)$ is an infinite field of characteristic $p\text{.}$

Let $D$ be an integral domain. Define a relation on $D$ by $a\sim b$ if $a$ and $b$ are associates in $D\text{.}$ Prove that $\sim$ is an equivalence relation on $D\text{.}$

Let $D$ be a Euclidean domain with Euclidean valuation $\nu \text{.}$ If $u$ is a unit in $D\text{,}$ show that $\nu (u)=\nu (1)\text{.}$

Let $D$ be a Euclidean domain with Euclidean valuation $\nu \text{.}$ If $a$ and $b$ are associates in $D\text{,}$ prove that $\nu (a)=\nu (b)\text{.}$

Show that $\mathbb{Z}[\sqrt{5}i]$ is not a unique factorization domain.

Prove or disprove: Every subdomain of a UFD is also a UFD.

An ideal of a commutative ring $R$ is said to be finitely generated if there exist elements $a_1,\dots ,a_n$ in $R$ such that every element $r$ in the ideal can be written as $a_1{r}_1+\cdots +a_n{r}_n$ for some $r_1,\dots ,r_n$ in $R\text{.}$ Prove that $R$ satisfies the ascending chain condition if and only if every ideal of $R$ is finitely generated.

Let $D$ be an integral domain with a descending chain of ideals $I_1\supset I_2\supset I_3\supset \cdots \text{.}$ Suppose that there exists an $N$ such that $I_k=I_N$ for all $k\ge N\text{.}$ A ring satisfying this condition is said to satisfy the descending chain condition, or DCC. Rings satisfying the DCC are called Artinian rings, after Emil Artin. Show that if $D$ satisfies the descending chain condition, it must satisfy the ascending chain condition.