18.4: Exercises

1

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

2

The Gaussian integers, ${\mathbb Z}[i]\text{,}$ are a UFD. Factor each of the following elements in ${\mathbb Z}[i]$ into a product of irreducibles.

1. $\displaystyle 5$
2. $\displaystyle 1 + 3i$
3. $\displaystyle 6 + 8i$
4. $\displaystyle 2$

3

Let $D$ be an integral domain.

1. Prove that $F_D$ is an abelian group under the operation of addition.
2. Show that the operation of multiplication is well-defined in the field of fractions, $F_D\text{.}$
3. Verify the associative and commutative properties for multiplication in $F_D\text{.}$

4

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

5

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

6

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

7

Let $F$ be a field.

1. Prove that the field of fractions of $F[x]\text{,}$ denoted by $F(x)\text{,}$ is isomorphic to the set all rational expressions $p(x) / q(x)\text{,}$ where $q(x)$ is not the zero polynomial.
2. Let $p(x_1, \ldots, x_n)$ and $q(x_1, \ldots, x_n)$ be polynomials in $F[x_1, \ldots, x_n]\text{.}$ Show that the set of all rational expressions $p(x_1, \ldots, x_n) / q(x_1, \ldots, x_n)$ is isomorphic to the field of fractions of $F[x_1, \ldots, x_n]\text{.}$ We denote the field of fractions of $F[x_1, \ldots, x_n]$ by $F(x_1, \ldots, x_n)\text{.}$

8

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 $p\text{.}$

9

Prove that the field of fractions of the Gaussian integers, ${\mathbb Z}[i]\text{,}$ is

$${\mathbb Q}(i) = \{ p + q i : p, q \in {\mathbb Q} \}\text{.} \nonumber$$

10

A field $F$ is called a prime field if it has no proper subfields. If $E$ is a subfield of $F$ and $E$ is a prime field, then $E$ is a prime subfield of $F\text{.}$

1. Prove that every field contains a unique prime subfield.
2. If $F$ is a field of characteristic 0, prove that the prime subfield of $F$ is isomorphic to the field of rational numbers, ${\mathbb Q}\text{.}$
3. If $F$ is a field of characteristic $p\text{,}$ prove that the prime subfield of $F$ is isomorphic to ${\mathbb Z}_p\text{.}$

11

Let ${\mathbb Z}[ \sqrt{2}\, ] = \{ a + b \sqrt{2} : a, b \in {\mathbb Z} \}\text{.}$

1. Prove that ${\mathbb Z}[ \sqrt{2}\, ]$ is an integral domain.
2. Find all of the units in ${\mathbb Z}[\sqrt{2}\, ]\text{.}$
3. Determine the field of fractions of ${\mathbb Z}[ \sqrt{2}\, ]\text{.}$
4. Prove that ${\mathbb Z}[ \sqrt{2} i ]$ is a Euclidean domain under the Euclidean valuation $\nu( a + b \sqrt{2}\, i) = a^2 + 2b^2\text{.}$

12

Let $D$ be a UFD. An element $d \in D$ is a greatest common divisor of $a$ and $b$ in $D$ if $d \mid a$ and $d \mid b$ and $d$ is divisible by any other element dividing both $a$ and $b\text{.}$

1. If $D$ is a PID and $a$ and $b$ are both nonzero elements of $D\text{,}$ prove there exists a unique greatest common divisor of $a$ and $b$ up to associates. That is, if $d$ and $d'$ are both greatest common divisors of $a$ and $b\text{,}$ then $d$ and $d'$ are associates. We write $\gcd( a, b)$ for the greatest common divisor of $a$ and $b\text{.}$
2. Let $D$ be a PID and $a$ and $b$ be nonzero elements of $D\text{.}$ Prove that there exist elements $s$ and $t$ in $D$ such that $\gcd(a, b) = as + bt\text{.}$

13

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 $D\text{.}$

14

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

15

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

16

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

17

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

18

An ideal of a commutative ring $R$ is said to be finitely generated if there exist elements $a_1, \ldots, a_n$ 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 $r_1, \ldots, r_n$ in $R\text{.}$ Prove that $R$ satisfies the ascending chain condition if and only if every ideal of $R$ is finitely generated.

19

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 $k \geq N\text{.}$ 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.

20

Let $R$ be a commutative ring with identity. We define a multiplicative subset of $R$ to be a subset $S$ such that $1 \in S$ and $ab \in S$ if $a, b \in S\text{.}$

1. Define a relation $\sim$ on $R \times S$ by $(a, s) \sim (a', s')$ if there exists an $s^\ast \in S$ such that $s^\ast(s' a -s a') = 0\text{.}$ Show that $\sim$ is an equivalence relation on $R \times S\text{.}$
2. Let $a/s$ denote the equivalence class of $(a,s) \in R \times S$ and let $S^{-1}R$ be the set of all equivalence classes with respect to $\sim\text{.}$ Define the operations of addition and multiplication on $S^{-1} R$ by

   $$\begin{align*} \frac{a}{s} + \frac{b}{t} & = \frac{at + b s}{s t}\\ \frac{a}{s} \frac{b}{t} & = \frac{a b}{s t}\text{,} \end{align*}$$

   respectively. Prove that these operations are well-defined on $S^{-1}R$ and that $S^{-1}R$ is a ring with identity under these operations. The ring $S^{-1}R$ is called the ring of quotients of $R$ with respect to $S\text{.}$

3. Show that the map $\psi : R \rightarrow S^{-1}R$ defined by $\psi(a) = a/1$ is a ring homomorphism.
4. If $R$ has no zero divisors and $0 \notin S\text{,}$ show that $\psi$ is one-to-one.
5. Prove that $P$ is a prime ideal of $R$ if and only if $S = R \setminus P$ is a multiplicative subset of $R\text{.}$
6. If $P$ is a prime ideal of $R$ and $S = R \setminus P\text{,}$ show that the ring of quotients $S^{-1}R$ has a unique maximal ideal. Any ring that has a unique maximal ideal is called a local ring.