Contemporary Abstract Algebra

Joseph Gallian

Chapter 13

Integral Domains - all with Video Answers

Educators

Chapter Questions

Problem 1

Verify that Examples 1 through 8 are as claimed.

Manik Pulyani

Manik Pulyani

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Problem 2

Which of Examples 1 through 5 are fields?

Jillian Rae Villa

Jillian Rae Villa

Numerade Educator

Problem 3

Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors.

James Kiss

Numerade Educator

Problem 4

List all zero-divisors in $Z_{20}$. Can you see a relationship between the zero-divisors of $Z_{20}$ and the units of $Z_{20}$ ?

Julie Silva

Numerade Educator

Problem 5

Show that every nonzero element of $Z_{n}$ is a unit or a zero-divisor.

James Chok

Numerade Educator

Problem 6

Find a nonzero element in a ring that is neither a zero-divisor nor a unit.

James Chok

Numerade Educator

Problem 7

Let $R$ be a finite commutative ring with unity. Prove that every nonzero element of $R$ is either a zero-divisor or a unit. What happens if we drop the "finite" condition on $R ?$

Kevin Harmer

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Problem 8

Let $a \neq 0$ belong to a commutative ring. Prove that $a$ is a zerodivisor if and only if $a^{2} b=0$ for some $b \neq 0$.

Willis James

Numerade Educator

Problem 9

Find elements $a, b$, and $c$ in the ring $Z \oplus Z \oplus Z$ such that $a b, a c$, and $b c$ are zero-divisors but $a b c$ is not a zero-divisor.

James Chok

Numerade Educator

Problem 10

Describe all zero-divisors and units of $Z \oplus Q \oplus Z$.

Uma Kumari

Numerade Educator

Problem 11

Let $d$ be an integer. Prove that $Z[\sqrt{d}]=\{a+b \sqrt{d} \mid a, b \in Z\}$ is an integral domain. (This exercise is referred to in Chapter 18.)

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 12

In $Z_{7}$, give a reasonable interpretation for the expressions $1 / 2$, $-2 / 3, \sqrt{-3}$, and $-1 / 6$.

Noah Sung

Numerade Educator

Problem 13

Give an example of a commutative ring without zero-divisors that is not an integral domain.

Amy Jiang

Numerade Educator

Problem 14

Find two elements $a$ and $b$ in a ring such that both $a$ and $b$ are zerodivisors, $a+b \neq 0$, and $a+b$ is not a zero-divisor.

James Chok

Numerade Educator

Problem 15

Let $a$ belong to a ring $R$ with unity and suppose that $a^{n}=0$ for some positive integer $n$. (Such an element is called nilpotent.) Prove that $1-a$ has a multiplicative inverse in $R$. [Hint: Consider $\left.(1-a)\left(1+a+a^{2}+\cdots+a^{n-1}\right) \cdot\right]$

Angelo Rendina

Numerade Educator

Problem 16

Show that the nilpotent elements of a commutative ring form a subring.

Shahab Ullah

Numerade Educator

Problem 17

Show that 0 is the only nilpotent element in an integral domain.

Joe Lesueur

Numerade Educator

Problem 18

A ring element $a$ is called an idempotent if $a^{2}=a$. Prove that the only idempotents in an integral domain are 0 and 1 .

Mohan Jain

Numerade Educator

Problem 19

Let $a$ and $b$ be idempotents in a commutative ring. Show that each of the following is also an idempotent: $a b, a-a b, a+b-a b$, $a+b-2 a b$

Mohan Jain

Numerade Educator

Problem 20

Show that $Z_{n}$ has a nonzero nilpotent element if and only if $n$ is divisible by the square of some prime.

Joe Lesueur

Numerade Educator

Problem 21

Let $R$ be the ring of real-valued continuous functions on $[-1,1]$. Show that $R$ has zero-divisors.

James Chok

Numerade Educator

Problem 22

Prove that if $a$ is a ring idempotent, then $a^{n}=a$ for all positive integers $n$.

Mohan Jain

Numerade Educator

Problem 23

Determine all ring elements that are both nilpotent elements and idempotents.

Shahab Ullah

Numerade Educator

Problem 24

Find a zero-divisor in $Z_{5}[i]=\left\{a+b i \mid a, b \in Z_{5}\right\}$.

Victor Salazar

Numerade Educator

Problem 25

Find an idempotent in $Z_{5}[i]=\left\{a+b i \mid a, b \in Z_{5}\right\}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

Find all units, zero-divisors, idempotents, and nilpotent elements in $Z_{3} \oplus Z_{6}$

AG

Ankit Gupta

Numerade Educator

Problem 27

Determine all elements of a ring that are both units and idempotents.

Ely Crowder

Numerade Educator

Problem 28

Let $R$ be the set of all real-valued functions defined for all real numbers under function addition and multiplication.
a. Determine all zero-divisors of $R$.
b. Determine all nilpotent elements of $R$.
c. Show that every nonzero element is a zero-divisor or a unit.

Anthony Ramos

Numerade Educator

Problem 29

(Subfield Test) Let $F$ be a field and let $K$ be a subset of $F$ with at least two elements. Prove that $K$ is a subfield of $F$ if, for any $a, b(b \neq 0)$ in $K, a-b$ and $a b^{-1}$ belong to $K$.

Gideon Idumah

Numerade Educator

Problem 30

Let $d$ be a positive integer. Prove that $Q[\sqrt{d}]=\{a+b \sqrt{d} \mid$ $a, b \in Q\}$ is a field.

Sandip Ranjan

Numerade Educator

Problem 31

Let $R$ be a ring with unity 1 . If the product of any pair of nonzero elements of $R$ is nonzero, prove that $a b=1$ implies $b a=1$.

Dushyant Barot

Numerade Educator

Problem 32

Let $R=\{0,2,4,6,8\}$ under addition and multiplication modulo
10. Prove that $R$ is a field.

Wendi Zhao

Numerade Educator

Problem 33

Formulate the appropriate definition of a subdomain (that is, a "sub" integral domain). Let $D$ be an integral domain with unity $1 .$ Show that $P=\{n \cdot 1 \mid n \in Z\}$ (that is, all integral multiples of 1 ) is a subdomain of $D$. Show that $P$ is contained in every subdomain of $D$. What can we say about the order of $P ?$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 34

Prove that there is no integral domain with exactly six elements. Can your argument be adapted to show that there is no integral domain with exactly four elements? What about 15 elements? Use these observations to guess a general result about the number of elements in a finite integral domain.

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 35

Let $F$ be a field of order $2^{n}$. Prove that char $F=2$.

John Nicolle

Numerade Educator

Problem 36

Determine all elements of an integral domain that are their own inverses under multiplication.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 37

Characterize those integral domains for which 1 is the only element that is its own multiplicative inverse.

Julie Silva

Numerade Educator

Problem 38

Determine all integers $n>1$ for which $(n-1) !$ is a zero-divisor in $Z_{n}$.

Julie Silva

Numerade Educator

Problem 39

Suppose that $a$ and $b$ belong to an integral domain.
a. If $a^{5}=b^{5}$ and $a^{3}=b^{3}$, prove that $a=b$.
b. If $a^{m}=b^{m}$ and $a^{n}=b^{n}$, where $m$ and $n$ are positive integers that are relatively prime, prove that $a=b$.

Trang Hoang

Numerade Educator

Problem 40

Find an example of an integral domain and distinct positive integers $m$ and $n$ such that $a^{m}=b^{m}$ and $a^{n}=b^{n}$, but $a \neq b$.

Sirat Shah

Numerade Educator

Problem 41

If $a$ is an idempotent in a commutative ring, show that $1-a$ is also an idempotent.

Mohan Jain

Numerade Educator

Problem 42

Construct a multiplication table for $Z_{2}[i]$, the ring of Gaussian integers modulo $2 .$ Is this ring a field? Is it an integral domain?

Vikash Ranjan

Numerade Educator

Problem 43

The nonzero elements of $Z_{3}[i]$ form an Abelian group of order 8 under multiplication. Is it isomorphic to $Z_{8}, Z_{4} \oplus Z_{2}$, or $Z_{2} \oplus Z_{2} \oplus Z_{2} ?$

Ely Crowder

Numerade Educator

Problem 44

Show that $Z_{7}[\sqrt{3}]=\left\{a+b \sqrt{3} \mid a, b \in Z_{7}\right\}$ is a field. For any positive integer $k$ and any prime $p$, determine a necessary and sufficient condition for $Z_{p}[\sqrt{k}]=\left\{a+b \sqrt{k} \mid a, b \in Z_{p}\right\}$ to be a field.

Gaurav Kalra

Numerade Educator

Problem 45

Show that a finite commutative ring with no zero-divisors and at least two elements has a unity.

James Kiss

Numerade Educator

Problem 46

Suppose that $a$ and $b$ belong to a commutative ring and $a b$ is a zero-divisor. Show that either $a$ or $b$ is a zero-divisor.

James Chok

Numerade Educator

Problem 47

Suppose that $R$ is a commutative ring without zero-divisors. Show that all the nonzero elements of $R$ have the same additive order.

James Kiss

Numerade Educator

Problem 48

Suppose that $R$ is a commutative ring without zero-divisors. Show that the characteristic of $R$ is 0 or prime.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 49

Let $x$ and $y$ belong to a commutative ring $R$ with prime characteristic $p$.
a. Show that $(x+y)^{p}=x^{p}+y^{p}$
b. Show that, for all positive integers $n,(x+y)^{p^{n}}=x^{p^{n}}+y^{p^{n}} .$
c. Find elements $x$ and $y$ in a ring of characteristic 4 such that $(x+y)^{4} \neq x^{4}+y^{4}$. (This exercise is referred to in Chapter 20.)

Aayush Gupta

Numerade Educator

Problem 50

Let $R$ be a commutative ring with unity 1 and prime characteristic. If $a \in R$ is nilpotent, prove that there is a positive integer $k$ such that $(1+a)^{k}=1$.

Nick Johnson

Numerade Educator

Problem 51

Show that any finite field has order $p^{n}$, where $p$ is a prime. Hint: Use facts about finite Abelian groups. (This exercise is referred to in Chapter 22.)

Brandon Collins

Brandon Collins

Numerade Educator

Problem 52

Give an example of an infinite integral domain that has characteristic $3 .$

Harshita Goel

Numerade Educator

Problem 53

Let $R$ be a ring and let $M_{2}(R)$ be the ring of $2 \times 2$ matrices with entries from $R$. Explain why these two rings have the same characteristic.

Nick Johnson

Numerade Educator

Problem 54

Let $R$ be a ring with $m$ elements. Show that the characteristic of $R$ divides $m$.

Sriparna Bhattacharjee

Sriparna Bhattacharjee

Numerade Educator

Problem 55

Explain why a finite ring must have a nonzero characteristic.

AG

Ankit Gupta

Numerade Educator

Problem 56

Find all solutions of $x^{2}-x+2=0$ over $Z_{3}[i] .$ (See Example 9.)

AG

Ankit Gupta

Numerade Educator

Problem 57

Consider the equation $x^{2}-5 x+6=0$.
a. How many solutions does this equation have in $Z_{7}$ ?
b. Find all solutions of this equation in $Z_{8}$.
c. Find all solutions of this equation in $Z_{12}$.
d. Find all solutions of this equation in $Z_{14}$.

Charles Carter

Numerade Educator

Problem 58

Find the characteristic of $Z_{4} \oplus 4 Z$.

Gopesh Vishwakarma

Gopesh Vishwakarma

Numerade Educator

Problem 59

Suppose that $R$ is an integral domain in which $20 \cdot 1=0$ and $12 \cdot 1=0$. (Recall that $n \cdot 1$ means the sum $1+1+\cdots+1$ with $n$ terms.) What is the characteristic of $R ?$

Doruk Isik

Numerade Educator

Problem 60

In a commutative ring of characteristic 2 , prove that the idempotents form a subring.

Vaibhav Jain

Numerade Educator

Problem 61

Describe the smallest subfield of the field of real numbers that contains $\sqrt{2}$. (That is, describe the subfield $K$ with the property that $K$ contains $\sqrt{2}$ and if $F$ is any subfield containing $\sqrt{2}$, then $F$ contains $K .$ )

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 62

Let $F$ be a finite field with $n$ elements. Prove that $x^{n-1}=1$ for all nonzero $x$ in $F$.

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 63

Let $F$ be a field of prime characteristic $p .$ Prove that $K=\{x \in F \mid$ $\left.x^{p}=x\right\}$ is a subfield of $F$.

Gideon Idumah

Numerade Educator

Problem 64

Suppose that $a$ and $b$ belong to a field of order 8 and that $a^{2}+a b+$ $b^{2}=0$. Prove that $a=0$ and $b=0$. Do the same when the field has order $2^{n}$ with $n$ odd.

Nick Johnson

Numerade Educator

Problem 65

Let $F$ be a field of characteristic 2 with more than two elements. Show that $(x+y)^{3} \neq x^{3}+y^{3}$ for some $x$ and $y$ in $F$.

Zachary Mitchell

Zachary Mitchell

Numerade Educator

Problem 66

Suppose that $F$ is a field with characteristic not 2 , and that the nonzero elements of $F$ form a cyclic group under multiplication. Prove that $F$ is finite.

Anthony Ramos

Numerade Educator

Problem 67

Suppose that $D$ is an integral domain and that $\phi$ is a nonconstant function from $D$ to the nonnegative integers such that $\phi(x y)=$ $\phi(x) \phi(y)$. If $x$ is a unit in $D$, show that $\phi(x)=1$.

Foster Wisusik

Numerade Educator

Problem 68

Let $F$ be a field of order 32 . Show that the only subfields of $F$ are $F$ itself and $\{0,1\}$.

Lucas Finney

Numerade Educator

Problem 69

Suppose that $F$ is a field with 27 elements. Show that for every element $a \in F, 5 a=-a$

John Nicolle

Numerade Educator

Problem 70

Let
$$
R=\left\{\left[\begin{array}{cc}
a & -b \\
b & a
\end{array}\right] \mid a, b \in Z_{7}\right\}
$$
with the usual matrix addition and multiplication and mod 7 addition and multiplication of the entries. Prove that $R$ is a commutative ring. How many elements are in $R ?$ Is $R$ a field? What happens when $Z_{7}$ is replaced by $Z_{5}$ ?

James Kiss

Numerade Educator