Contemporary Abstract Algebra

Joseph Gallian

Chapter 16 Polynomial Rings - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $f(x)=4 x^{3}+2 x^{2}+x+3$ and $g(x)=3 x^{4}+3 x^{3}+3 x^{2}+x+4$,
where $f(x), g(x) \in Z_{5}[x] .$ Compute $f(x)+g(x)$ and $f(x) \cdot g(x)$.

Rylie Howey

Numerade Educator

07:23

Problem 2

In $Z_{3}[x]$, show that the distinct polynomials $x^{4}+x$ and $x^{2}+x$ determine the same function from $Z_{3}$ to $Z_{3}$.

Shuyang Fu

Numerade Educator

00:50

Problem 3

Show that $x^{2}+3 x+2$ has four zeros in $Z_{6}$.

Ankit Gupta

Numerade Educator

01:28

Problem 4

If $R$ is a commutative ring, show that the characteristic of $R[x]$ is the same as the characteristic of $R$.

Chris Trentman

Numerade Educator

05:55

Problem 5

Prove Corollary 1 of Theorem $16.2 .$

Madi Sousa

Numerade Educator

01:56

Problem 6

List all the polynomials of degree 2 in $Z_{2}[x] .$ Which of these are equal as functions from $Z_{2}$ to $Z_{2}$ ?

Diogo Caetano

Numerade Educator

01:12

Problem 7

Find two distinct cubic polynomials over $Z_{2}$ that determine the same function from $Z_{2}$ to $Z_{2}$.

Chris Trentman

Numerade Educator

00:27

Problem 8

For any positive integer $n$, how many polynomials are there of degree $n$ over $Z_{2}$ ? How many distinct polynomial functions from $Z_{2}$ to $Z_{2}$ are there?

Ankit Gupta

Numerade Educator

00:32

Problem 9

Let $f(x)=5 x^{4}+3 x^{3}+1$ and $g(x)=3 x^{2}+2 x+1$ in $Z_{7}[x]$. Determine the quotient and remainder upon dividing $f(x)$ by $g(x)$.

James Kiss

Numerade Educator

03:58

Problem 10

Let $R$ be a commutative ring. Show that $R[x]$ has a subring isomorphic to $R$.

Anthony Ramos

Numerade Educator

01:34

If $\phi: R \rightarrow S$ is a ring homomorphism, define $\bar{\phi}: R[x] \rightarrow S[x]$ by $\left(a_{n} x^{n}\right.$ $\left.+\cdots+a_{0}\right) \rightarrow \phi\left(a_{n}\right) x^{n}+\cdots+\phi\left(a_{0}\right) .$ Show that $\bar{\phi}$ is a ring homomorphism. (This exercise is referred to in Chapter $33 .$ )

Manik Pulyani

Numerade Educator

03:58

Problem 12

If the rings $R$ and $S$ are isomorphic, show that $R[x]$ and $S[x]$ are isomorphic. (The converse to not true-see $[1] .)$

Anthony Ramos

Numerade Educator

05:55

Problem 13

Prove Corollary 2 of Theorem $16.2$.

Madi Sousa

Numerade Educator

02:51

Problem 14

Let $f(x)$ and $g(x)$ be cubic polynomials with integer coefficients such that $f(a)=g(a)$ for four integer values of $a$. Prove that $f(x)=$ $g(x) .$ Generalize.

Kenwa Nandi

Numerade Educator

01:00

Problem 15

Show that the polynomial $2 x+1$ in $Z_{4}[x]$ has a multiplicative inverse in $Z_{4}[x]$.

Lucas Finney

Numerade Educator

01:56

Problem 16

Are there any nonconstant polynomials in $Z[x]$ that have multiplicative inverses? Explain your answer.

Diogo Caetano

Numerade Educator

07:30

Problem 17

Let $p$ be a prime. Are there any nonconstant polynomials in $Z_{p}[x]$ that have multiplicative inverses? Explain your answer.

Lucas Finney

Numerade Educator

00:28

Problem 18

Show that Theorem $16.4$ is false for any commutative ring that has a zero divisor.

James Kiss

Numerade Educator

01:35

Problem 19

(Degree Rule) Let $D$ be an integral domain and $f(x), g(x) \in D[x]$. Prove that deg $(f(x) \cdot g(x))=\operatorname{deg} f(x)+\operatorname{deg} g(x) .$ Show, by example, that for commutative ring $R$ it is possible that $\operatorname{deg} f(x) g(x)<$ $\operatorname{deg} f(x)+\operatorname{deg} g(x)$, where $f(x)$ and $g(x)$ are nonzero elements in $R[x]$. (This exercise is referred to in this chapter, Chapter 17 , and Chapter 18.)

Sriram Soundarrajan

Numerade Educator

01:39

Problem 20

Prove that the ideal $\langle x\rangle$ in $Q[x]$ is maximal.

Mohamed Mohamed

Numerade Educator

03:12

Problem 21

Let $f(x)$ belong to $F[x]$, where $F$ is a field. Let $a$ be a zero of $f(x)$ of multiplicity $n$, and write $f(x)=(x-a)^{n} q(x)$. If $b \neq a$ is a zero of $q(x)$, show that $b$ has the same multiplicity as a zero of $q(x)$ as it does for $f(x)$. (This exercise is referred to in this chapter.)

Joshua Roe

Numerade Educator

01:34

Problem 22

Prove that for any positive integer $n$, a field $F$ can have at most a finite number of elements of multiplicative order at most $n$.

Clarissa Noh

Numerade Educator

02:38

Problem 23

Let $F$ be a field, and let $f(x)$ and $g(x)$ belong to $F[x]$. If there is no polynomial of positive degree in $F[x]$ that divides both $f(x)$ and $g(x)$ [in this case, $f(x)$ and $g(x)$ are said to be relatively prime $]$, prove that there exist polynomials $h(x)$ and $k(x)$ in $F[x]$ with the property that $f(x) h(x)+g(x) k(x)=1 .$ (This exercise is referred to in Chapter 20.)

Vikash Ranjan

Numerade Educator

00:38

Problem 24

Let $F$ be an infinite field and let $f(x), g(x) \in F[x]$. If $f(a)=g(a)$ for infinitely many elements $a$ of $F$, show that $f(x)=g(x)$.

Hast Aggarwal

Numerade Educator

03:27

Problem 25

Let $F$ be a field and let $p(x) \in F[x] .$ If $f(x), g(x) \in F[x]$ and $\operatorname{deg} f(x)<\operatorname{deg} p(x)$ and deg $g(x)<\operatorname{deg} p(x)$, show that $f(x)+$ $\langle p(x)\rangle=g(x)+\langle p(x)\rangle$ implies $f(x)=g(x) .$ (This exercise is referred to in Chapter 20.)

Lucas Finney

Numerade Educator

00:50

Problem 26

Prove that $Z[x]$ is not a principal ideal domain. (Compare this with Theorem 16.3.)

Hasan Saifee

Numerade Educator

02:19

Problem 27

Find a polynomial with integer coefficients that has $1 / 2$ and $-1 / 3$ as zeros.

Vishal Parmar

Numerade Educator

10:11

Problem 28

Let $f(x) \in \mathbf{R}[x] .$ Suppose that $f(a)=0$ but $f^{\prime}(a) \neq 0$, where $f^{\prime}(x)$ is the derivative of $f(x)$. Show that $a$ is a zero of $f(x)$ of multiplicity 1 .

Anthony Ramos

Numerade Educator

01:02

Problem 29

Show that Corollary 2 of Theorem $16.2$ is true over any commutative ring with unity.

Linh Vu

Numerade Educator

View

Problem 30

Show that Theorem $16.4$ is true for polynomials over integral domains.

Nick Johnson

Numerade Educator

05:20

Problem 31

Let $F$ be a field and let
$$
\begin{aligned}
&I=\left\{a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \mid a_{n}, a_{n-1}, \ldots, a_{0} \in F\right. \text { and } \\
&\left.\quad a_{n}+a_{n-1}+\cdots+a_{0}=0\right\}
\end{aligned}
$$
Show that $I$ is an ideal of $F[x]$ and find a generator for $I$.

Clarissa Noh

Numerade Educator

02:26

Problem 32

Let $F$ be a field and let $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \in F[x]$. Prove that $x-1$ is a factor of $f(x)$ if and only if $a_{n}+a_{n-1}+\cdots+$ $a_{0}=0$.

Runpeng Li

Numerade Educator

01:00

Problem 33

Let $m$ be a fixed positive integer. For any integer $a$, let $\bar{a}$ denote $a \bmod m$. Show that the mapping of $\phi: Z[x] \rightarrow Z_{m}[x]$ given by $$\phi\left(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\right)=\bar{a}_{n} x^{n}+\bar{a}_{n-1} x^{n-1}+\cdots+\bar{a}_{0}$$
is a ring homomorphism. (This exercise is referred to in Chapters 17 and $33 .$ )

James Chok

Numerade Educator

01:17

Problem 34

Find infinitely many polynomials $f(x)$ in $Z_{3}[x]$ such that $f(a)=0$ for all $a$ in $Z_{3}$.

Aman Gupta

Numerade Educator

01:59

Problem 35

For every prime $p$, show that
$$x^{p-1}-1=(x-1)(x-2) \cdots[x-(p-1)]$$
in $Z_{p}[x]$.

James Chok

Numerade Educator

08:50

Problem 36

Let $\phi$ be the ring homomorphism from $Z[x]$ to $Z$ given by $\phi(f(x))=$ $f(1)$. Find a polynomial $g(x)$ in $Z[x]$ such that Ker $\phi=\langle g(x)\rangle$. Is there more than one possibility for $g(x)$ ? To what familiar ring is $Z[x] / \operatorname{Ker} \phi$ isomorphic? Do this exercise with $Z$ replaced by $Q$.

Ely Crowder

Numerade Educator

00:32

Problem 37

Give an example of a field that properly contains the field of complex numbers $\mathbf{C}$.

James Kiss

Numerade Educator

05:06

Problem 38

(Wilson's Theorem) For every integer $n>1$, prove that $(n-1) !$ $\bmod n=n-1$ if and only if $n$ is prime.

Chris Trentman

Numerade Educator

08:41

Problem 39

For every prime $p$, show that $(p-2) ! \bmod p=1$.

Mengchun Cai

Numerade Educator

00:54

Problem 40

Find the remainder upon dividing $98 !$ by 101 .

Kumareshwaran Rathinasabapathy

Numerade Educator

01:19

Problem 41

Prove that $(50 !)^{2} \bmod 101=-1$ mod 101 .

Manik Pulyani

Numerade Educator

03:07

Problem 42

If $I$ is an ideal of a ring $R$, prove that $I[x]$ is an ideal of $R[x]$.

Kevin Harmer

Numerade Educator

00:32

Problem 43

Give an example of a commutative ring $R$ with unity and a maximal ideal $I$ of $R$ such that $I[x]$ is not a maximal ideal of $R[x]$

Ian Maurer

Numerade Educator

03:56

Problem 44

Let $R$ be a commutative ring with unity. If $I$ is a prime ideal of $R$, prove that $I[x]$ is a prime ideal of $R[x]$.

Sriparna Bhattacharjee

Numerade Educator

01:16

Problem 45

Let $F$ be an infinite field and let $f(x) \in F[x] .$ If $f(a)=0$ for infinitely many elements $a$ of $F$, show that $f(x)=0$.

Bahar Tehranipoor

Numerade Educator

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Problem 46

Prove that $Q[x] /\left\langle x^{2}-2\right\rangle$ is ring-isomorphic to $Q[\sqrt{2}]=\{a+$ $b \sqrt{2} \mid a, b \in Q\}$

Nick Johnson

Numerade Educator

01:29

Problem 47

Let $f(x) \in \mathbf{R}[x] .$ If $f(a)=0$ and $f^{\prime}(a)=0\left[f^{\prime}(a)\right.$ is the derivative of $f(x)$ at $a]$, show that $(x-a)^{2}$ divides $f(x)$.

Adrian Co

Numerade Educator

03:37

Problem 48

Let $F$ be a field and let $I=\{f(x) \in F[x] \mid f(a)=0$ for all $a$ in $F\}$. Prove that $I$ is an ideal in $F[x]$. Prove that $I$ is infinite when $F$ is finite and $I=\{0\}$ when $F$ is infinite. When $F$ is finite, find a monic polynomial $g(x)$ such that $I=\langle g(x)\rangle$.

J Hardin

Numerade Educator

04:32

Problem 49

Let $g(x)$ and $h(x)$ belong to $Z[x]$ and let $h(x)$ be monic. If $h(x)$ divides $g(x)$ in $Q[x]$, show that $h(x)$ divides $g(x)$ in $Z[x]$. (This exercise is referred to in Chapter $33 .$.)

Sandip Ranjan

Numerade Educator

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Problem 50

Let $\mathrm{R}$ be a ring and $x$ be an indeterminate. Prove that the rings $R[x]$ and $R\left[x^{2}\right]$ are ring-isomorphic.

Nick Johnson

Numerade Educator

00:51

Problem 51

Let $f(x)$ be a nonconstant element of $Z[x]$. Prove that $f(x)$ takes on infinitely many values in $Z$.

Mohamed Mohamed

Numerade Educator

01:57

Problem 52

Let $f(x)$ be a nonconstant element in $Z[x]$. Prove that $\langle f(x)\rangle$ is not maximal in $Z[x]$

Richard Foote

Numerade Educator

03:58

Problem 53

Suppose that $F$ is a field and there is a ring homomorphism from $Z$ onto $F$. Show that $F$ is isomorphic to $Z_{p}$ for some prime $p$.

Anthony Ramos

Numerade Educator

04:32

Problem 54

Let $f(x)$ belong to $Z_{p}[x]$. Prove that if $f(b)=0$, then $f\left(b^{p}\right)=0$.

Sandip Ranjan

Numerade Educator

03:15

Problem 55

Suppose $f(x)$ is a polynomial with odd integer coefficients and even degree. Prove that $f(x)$ has no rational zeros.

Ankit Gupta

Numerade Educator

00:57

Problem 56

Find the remainder when $x^{51}$ is divided by $x+4$ in $Z_{7}[x]$.

Carson Merrill

Numerade Educator

01:41

Problem 57

Let $F$ be a field. Show that there exist $a, b \in F$ with the property that $x^{2}+x+1$ divides $x^{43}+a x+b$.

Zachary Mitchell

Numerade Educator

03:00

Problem 58

Let $f(x)=a_{m} x^{m}+a_{m-1} x^{m-1}+\cdots+a_{0}$ and $g(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+$
$\cdots+b_{0}$ belong to $Q[x]$ and suppose that $f(x) g(x)$ belongs to $Z[x] .$ Prove that $a_{i} b_{j}$ is an integer for every $i$ and $j .$

Srilakshmi E K

Numerade Educator

10:24

Problem 59

Let $f(x)$ belong to $Z[x]$. If $a \bmod m=b \bmod m$, prove that $f(a)$ mod $m=f(b) \bmod m .$ Prove that if both $f(0)$ and $f(1)$ are odd, then $f$ has no zero in $Z$.

Heather Zimmers

Numerade Educator

01:49

Problem 60

For any field $F$, recall that $F(x)$ denotes the field of quotients of the ring $F[x]$. Prove that there is no element in $F(x)$ whose square is $x$.

Will Erickson

Numerade Educator

01:14

Problem 61

Show that 1 is the only solution of $x^{25}-1=0$ in $Z_{37}$.

Helen Latting

Numerade Educator

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