## Exercises18.4Exercises

###### 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

\begin{equation*} {\mathbb Q}(i) = \{ p + q i : p, q \in {\mathbb Q} \}\text{.} \end{equation*}
###### 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.