Contemporary Abstract Algebra

Joseph Gallian

Chapter 18 Divisibility in Integral Domains - all with Video Answers

Educators

Chapter Questions

03:59

Problem 1

For the ring $Z[\sqrt{d}]=\{a+b \sqrt{d} \mid a, b \in Z\}$, where $d \neq 1$ and $d$ is not divisible by the square of a prime, prove that the norm $N(a+$ $b \sqrt{d})=\left|a^{2}-d b^{2}\right|$ satisfies the four assertions made preceding Example 1 . (This exercise is referred to in this chapter.)

Aayush Gupta

Numerade Educator

00:58

Problem 2

In an integral domain, show that $a$ and $b$ are associates if and only if $\langle a\rangle=\langle b\rangle .$

Varsha Aggarwal

Numerade Educator

03:13

Problem 3

Show that the union of a chain $I_{1} \subset I_{2} \subset \cdots$ of ideals of a ring $R$ is an ideal of $R$. (This exercise is referred to in this chapter.)

Gideon Idumah

Numerade Educator

01:29

Problem 4

In an integral domain, show that the product of an irreducible and a unit is an irreducible.

Chris Trentman

Numerade Educator

00:41

Problem 5

Suppose that $a$ and $b$ belong to an integral domain and $b \neq 0$. Show that $\langle a b\rangle$ is a proper subset of $\langle b\rangle$ if and only if $a$ is not a unit. This exercise is referred to in this chapter.

Varsha Aggarwal

Numerade Educator

12:54

Problem 6

Let $D$ be an integral domain. Define $a \sim b$ if $a$ and $b$ are associates. Show that this defines an equivalence relation on $D$.

Chris Trentman

Numerade Educator

00:34

Problem 7

In the notation of Example 7 , show that $d(x y)=d(x) d(y)$.

Christopher Stanley

Numerade Educator

04:33

Problem 8

Let $D$ be a Euclidean domain with measure $d$. Prove that $u$ is a unit in $D$ if and only if $d(u)=d(1)$.

James Chok

Numerade Educator

01:37

Problem 9

Let $D$ be a Euclidean domain with measure $d$. Show that if $a$ and $b$ are associates in $D$, then $d(a)=d(b)$.

James Chok

Numerade Educator

05:03

Problem 10

Let $D$ be a principal ideal domain and let $p \in D$. Prove that $\langle p\rangle$ is a maximal ideal in $D$ if and only if $p$ is irreducible.

Varsha Aggarwal

Numerade Educator

03:16

Problem 11

Trace through the argument given in Example 7 to find $q$ and $r$ in $Z[i]$ such that $3-4 i=(2+5 i) q+r$ and $d(r)<d(2+5 i)$

Shaza Hammoud

Numerade Educator

01:08

Problem 12

Let $D$ be a principal ideal domain. Show that every proper ideal of $D$ is contained in a maximal ideal of $D$.

Linda Hand

Numerade Educator

01:44

Problem 13

In $Z[\sqrt{-5}]$, show that 21 does not factor uniquely as a product of irreducibles.

Julie Silva

Numerade Educator

04:09

Problem 14

Show that $1-i$ is an irreducible in $Z[i]$.

Narayan Hari

Numerade Educator

06:11

Problem 15

Show that $Z[\sqrt{-6}]$ is not a unique factorization domain. (Hint:
Factor 10 in two ways.) Why does this show that $Z[\sqrt{-6}]$ is not a principal ideal domain?

Linnea Reyes-Lamon

Numerade Educator

01:18

Problem 16

Give an example of a unique factorization domain with a subdomain that does not have a unique factorization.

Yujie Wang

College of San Mateo

04:09

Problem 17

In $Z[i]$, show that 3 is irreducible but 2 and 5 are not.

Narayan Hari

Numerade Educator

04:51

Problem 18

Prove that 7 is irreducible in $Z[\sqrt{6}]$, even though $N(7)$ is not prime.

Aayush Gupta

Numerade Educator

00:57

Problem 19

Prove that if $p$ is a prime in $Z$ that can be written in the form $a^{2}+b^{2}$, then $a+b i$ is irreducible in $Z[i]$. Find three primes that have this property and the corresponding irreducibles.

Trang Hoang

Numerade Educator

00:50

Problem 20

Prove that $Z[\sqrt{-3}]$ is not a principal ideal domain.

Hasan Saifee

Numerade Educator

03:59

Problem 21

In $Z[\sqrt{-5}]$, prove that $1+3 \sqrt{-5}$ is irreducible but not prime.

Mohamed Mohamed

Numerade Educator

03:59

Problem 22

In $Z[\sqrt{5}]$, prove that both 2 and $1+\sqrt{5}$ are irreducible but not prime.

Mohamed Mohamed

Numerade Educator

03:59

Problem 23

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

Mohamed Mohamed

Numerade Educator

03:23

Problem 24

Let $F$ be a field. Show that in $F[x]$ a prime ideal is a maximal ideal.

Anurag Kumar

Numerade Educator

04:29

Problem 25

Let $d$ be an integer less than $-1$ that is not divisible by the square of a prime. Prove that the only units of $Z[\sqrt{d}]$ are $+1$ and $-1$.

Trang Hoang

Numerade Educator

09:43

Problem 26

In $Z[\sqrt{2}]=\{a+b \sqrt{2} \mid a, b \in Z\}$, show that every element of the form $(3+2 \sqrt{2})^{n}$ is a unit, where $n$ is a positive integer.

Sandip Ranjan

Numerade Educator

01:35

Problem 27

If $a$ and $b$ belong to $Z[\sqrt{d}]$, where $d$ is not divisible by the square of a prime and $a b$ is a unit, prove that $a$ and $b$ are units.

James Chok

Numerade Educator

02:22

Problem 28

For a commutative ring with unity we may define associates, irreducibles, and primes exactly as we did for integral domains. With these definitions, show that both 2 and 3 are prime in $Z_{12}$ but 2 is irreducible and 3 is not.

James Chok

Numerade Educator

18:04

Problem 29

Let $n$ be a positive integer and $p$ a prime that divides $n$. Prove that $p$ is prime in $Z_{n^{*}}$ (See Exercise 28).

Oswaldo Jiménez

Numerade Educator

03:56

Problem 30

Let $p$ be a prime divisor of a positive integer $n$. Prove that $p$ is irreducible in $Z_{n}$ if and only if $p^{2}$ divides $n$. (See Exercise 28).

Sriparna Bhattacharjee

Numerade Educator

01:03

Problem 31

Prove or disprove that if $D$ is a principal ideal domain, then $D[x]$ is a principal ideal domain.

Raj Bala

Numerade Educator

01:28

Problem 32

Determine the units in $Z[i]$.

Manik Pulyani

Numerade Educator

08:41

Problem 33

Let $p$ be a prime in an integral domain. If $p \mid a_{1} a_{2} \cdots a_{n}$, prove that $p$ divides some $a_{i}$. (This exercise is referred to in this chapter.)

Mengchun Cai

Numerade Educator

04:55

Problem 34

Show that $3 x^{2}+4 x+3 \in Z_{5}[x]$ factors as $(3 x+2)(x+4)$ and $(4 x+1)(2 x+3)$. Explain why this does not contradict the corollary of Theorem $18.3$.

Stephanie Carter

Numerade Educator

00:57

Problem 35

Let $D$ be a principal ideal domain and $p$ an irreducible element of $D$. Prove that $D /\langle p\rangle$ is a field.

Trang Hoang

Numerade Educator

02:33

Problem 36

Show that an integral domain with the property that every strictly decreasing chain of ideals $I_{1} \supset I_{2} \supset \cdots$ must be finite in length is a field.

Nick Johnson

Numerade Educator

04:52

Problem 37

An ideal $A$ of a commutative ring $R$ with unity is said to be finitely generated if there exist elements $a_{1}, a_{2}, \ldots, a_{n}$ of $A$ such that $A=\left\langle a_{1}, a_{2}, \ldots, a_{n}\right\rangle .$ An integral domain $R$ is said to satisfy the ascending chain condition if every strictly increasing chain of ideals $I_{1} \subset I_{2} \subset \cdots$ must be finite in length. Show that an integral domain $R$ satisfies the ascending chain condition if and only if every ideal of $R$ is finitely generated.

Bobby Barnes

University of North Texas

01:03

Problem 38

Prove or disprove that a subdomain of a Euclidean domain is a Euclidean domain.

Raj Bala

Numerade Educator

02:04

Problem 39

Show that for any nontrivial ideal $I$ of $Z[i], Z[i] / I$ is finite.

Chris Trentman

Numerade Educator

01:01

Problem 40

Find the inverse of $1+\sqrt{2}$ in $Z[\sqrt{2}]$. What is the multiplicative order of $1+\sqrt{2}$ ?

Ankit Gupta

Numerade Educator

00:20

Problem 41

In $Z[\sqrt{-7}]$, show that $N(6+2 \sqrt{-7})=N(1+3 \sqrt{-7})$ but $6+$ $2 \sqrt{-7}$ and $1+3 \sqrt{-7}$ are not associates.

Vivek Kumar

Numerade Educator

02:36

Problem 42

Let $R=Z \oplus Z \oplus \cdots$ (the collection of all sequences of integers under componentwise addition and multiplication). Show that $R$ has ideals $I_{1}, I_{2}, I_{3}, \ldots$ with the property that $I_{1} \subset I_{2} \subset I_{3} \subset \cdots$. (Thus $R$ does not have the ascending chain condition.)

James Chok

Numerade Educator

01:59

Problem 43

Prove that in a unique factorization domain, an element is irreducible if and only if it is prime.

James Chok

Numerade Educator

02:14

Problem 44

Let $F$ be a field and let $R$ be the integral domain in $F[x]$ generated by $x^{2}$ and $x^{3}$. (That is, $R$ is contained in every integral domain in $F[x]$ that contains $x^{2}$ and $x^{3}$.) Show that $R$ is not a unique factorization domain.