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Section18.3Exercises

ΒΆ
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $-1\\text{.}$

Hint

Note that $z^{-1} = 1/(a + b\\sqrt{3}\\, i) = (a -b \\sqrt{3}\\, i)/(a^2 + 3b^2)$ is in ${\\mathbb Z}[\\sqrt{3}\\, i]$ if and only if $a^2 + 3 b^2 = 1\\text{.}$ The only integer solutions to the equation are $a = \\pm 1, b = 0\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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. 5

  2. $1 + 3i$

  3. $6 + 8i$

  4. 2

Hint

(a) $5 = -i(1 + 2i)(2 + i)\\text{;}$ (c) $6 + 8i = -i(1 + i)^2(2 + i)^2\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $F_D\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
4

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

Hint

True.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
5

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
6

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $F(x_1, \\ldots, x_n)\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $p\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
9

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

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

Let $z = a + bi$ and $w = c + di \\neq 0$ be in ${\\mathbb Z}[i]\\text{.}$ Prove that $z/w \\in {\\mathbb Q}(i)\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 ${\\mathbb Z}_p\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $\\nu( a + b \\sqrt{2}\\, i) = a^2 + 2b^2\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $\\gcd(a, b) = as + bt\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $D\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
14

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
15

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

Hint

Let $a = ub$ with $u$ a unit. Then $\\nu(b) \\leq \\nu(ub) \\leq \\nu(a)\\text{.}$ Similarly, $\\nu(a) \\leq \\nu(b)\\text{.}$

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
16

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

Hint

Show that 21 can be factored in two different ways.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
"]}, {"cell_type":"markdown", "metadata":{}, "source":["
17

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

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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 $r \\in R$ 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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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.

"]}, {"cell_type":"markdown", "metadata":{}, "source":["
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

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

    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.

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