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Contemporary Abstract Algebra

Joseph Gallian

Chapter 22

Finite Fields - all with Video Answers

Educators

Chapter Questions

03:03

Problem 1

Find $[\mathrm{GF}(729): \mathrm{GF}(9)]$ and $[\mathrm{GF}(64): \mathrm{GF}(8)]$.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
03:56

Problem 2

If $m$ divides $n$, show that $\left[\mathrm{GF}\left(p^{n}\right): \mathrm{GF}\left(p^{m}\right)\right]=n / m$.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:11

Problem 3

Draw the lattice of subfields of $\mathrm{GF}(64)$.

Aadit Sharma
Aadit Sharma
Numerade Educator
04:12

Problem 4

Let $\alpha$ be a zero of $x^{3}+x^{2}+1$ in some extension field of $Z_{2}$. Find the multiplicative inverse of $\alpha+1$ in $Z_{2}[\alpha]$.

Joshua Sieverding
Joshua Sieverding
Numerade Educator
02:19

Problem 5

Let $\alpha$ be a zero of $x^{3}+x^{2}+1$ in some extension field of $Z_{2}$. Solve the equation $(\alpha+1) x+\alpha=\alpha^{2}+1$ for $x$.

Ekaveera Kumar
Ekaveera Kumar
Numerade Educator
06:44

Problem 6

Prove that every non-identity element in $\mathrm{GF}(32)^{*}$ is a generator of $\mathrm{GF}(32)^{*}$

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
09:32

Problem 7

Let $\alpha$ be a zero of $f(x)=x^{2}+2 x+2$ in some extension field of $Z_{3}$. Find the other zero of $f(x)$ in $Z_{3}[\alpha]$.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
01:07

Problem 8

Let $\alpha$ be a zero of $f(x)=x^{3}+x+1$ in some extension field of $Z_{2}$. Find the other zeros of $f(x)$ in $Z_{2}[\alpha]$.

AG
Ankit Gupta
Numerade Educator
03:13

Problem 9

Let $K$ be a finite extension field of a finite field $F$. Show that there is an element $a$ in $K$ such that $K=F(a)$.

Gideon Idumah
Gideon Idumah
Numerade Educator
01:27

Problem 10

How many elements of the cyclic group $\mathrm{GF}(81)^{*}$ are generators?

Anthony Ramos
Anthony Ramos
Numerade Educator
02:49

Problem 11

Let $f(x)$ be a cubic irreducible over $Z_{2} .$ Prove that the splitting field of $f(x)$ over $Z_{2}$ has order 8 .

Lucas Finney
Lucas Finney
Numerade Educator
03:58

Problem 12

Prove that the rings $Z_{3}[x] /\left\langle x^{2}+x+2\right\rangle$ and $Z_{3}[x] /\left\langle x^{2}+2 x+2\right\rangle$ are isomorphic.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:19

Problem 13

Show that the Frobenius mapping $\phi: \mathrm{GF}\left(p^{n}\right) \rightarrow \mathrm{GF}\left(p^{n}\right)$, given by $a \rightarrow a^{p}$, is a ring automorphism of order $n$ (that is, $\phi^{n}$ is the identity mapping). (This exercise is referred to in Chapter $32 .$ )

Anthony Ramos
Anthony Ramos
Numerade Educator
00:38

Problem 14

Determine the possible finite fields whose largest proper subfield is $\mathrm{GF}\left(2^{5}\right)$

ES
Esraa Samir
Numerade Educator
01:06

Problem 15

Prove that the degree of any irreducible factor of $x^{8}-x$ over $Z_{2}$ is 1 or 3 .

Carson Merrill
Carson Merrill
Numerade Educator
02:18

Problem 16

Find the smallest field that has exactly 6 subfields.

Matt Just
Matt Just
Numerade Educator
01:22

Problem 17

Find the smallest field of characteristic 2 that contains an element whose multiplicative order is 5 and the smallest field of characteristic 3 that contains an element whose multiplicative order is 5 .

Harshita Goel
Harshita Goel
Numerade Educator
03:04

Problem 18

Verify that the factorization for $f(x)=x^{3}+x^{2}+1$ over $Z_{2}$ given in Example 2 is correct by expanding.

Jacquelyn Calos
Jacquelyn Calos
Numerade Educator
01:40

Problem 19

Show that $x$ is a generator of the cyclic group $\left(Z_{3}[x] /\left\langle x^{3}+2 x+1\right\rangle\right)^{*}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:23

Problem 20

Suppose that $f(x)$ is a fifth-degree polynomial that is irreducible over $Z_{2}$. Prove that $x$ is a generator of the cyclic group $\left(Z_{2}[x] /\langle f(x)\rangle\right)^{*}$

Gregory Higby
Gregory Higby
Numerade Educator
01:40

Problem 21

Show that $x$ is not a generator of the cyclic group $\left(Z_{3}[x] /\left\langle x^{3}+\right.\right.$ $2 x+2\rangle)^{*}$. Find one such generator.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
07:23

Problem 22

If $f(x)$ is a cubic irreducible polynomial over $Z_{3}$, prove that either $x$ or $2 x$ is a generator for the cyclic group $\left(Z_{3}[x] /\langle f(x)\rangle\right)^{*}$.

Shuyang Fu
Shuyang Fu
Numerade Educator
05:55

Problem 23

Prove the uniqueness portion of Theorem $22.3$ using a group theoretic argument.

Madi Sousa
Madi Sousa
Numerade Educator
01:30

Problem 24

Suppose that $\alpha$ and $\beta$ belong to $\mathrm{GF}(81)^{*}$, with $|\alpha|=5$ and $|\beta|=16$. Show that $\alpha \beta$ is a generator of $\mathrm{GF}(81)^{*}$.

Vysakh M
Vysakh M
Numerade Educator
01:10

Problem 25

Construct a field of order 9 and carry out the analysis as in Example 1, including the conversion table.

Carson Merrill
Carson Merrill
Numerade Educator
01:35

Problem 26

Show that any finite subgroup of the multiplicative group of a field is cyclic.

Nick Johnson
Nick Johnson
Numerade Educator
03:13

Problem 27

Show that the set $K$ in the proof of Theorem $22.3$ is a subfield.

Gideon Idumah
Gideon Idumah
Numerade Educator
03:56

Problem 28

If $g(x)$ is irreducible over $\mathrm{GF}(p)$ and $g(x)$ divides $x^{p^{n}}-x$, prove that deg $g(x)$ divides $n$.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:57

Problem 29

Use a purely group theoretic argument to show that if $F$ is a field of order $p^{n}$, then every element of $F^{*}$ is a zero of $x^{p^{n}}-x$. (This exercise is referred to in the proof of Theorem 22.1.)

Wendi Zhao
Wendi Zhao
Numerade Educator
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Problem 30

Draw the subfield lattices of $\mathrm{GF}\left(3^{18}\right)$ and of $\mathrm{GF}\left(2^{30}\right)$.

Dan Ni
Dan Ni
Numerade Educator
05:36

Problem 31

How does the subfield lattice of $\mathrm{GF}\left(2^{30}\right)$ compare with the subfield lattice of $\mathrm{GF}\left(3^{30}\right)$ ?

Daniel Kyinakwa
Daniel Kyinakwa
Numerade Educator
03:12

Problem 32

If $p(x)$ is a polynomial in $Z_{p}[x]$ with no multiple zeros, show that $p(x)$ divides $x^{p^{n}}-x$ for some $n$

Karly Williams
Karly Williams
Numerade Educator
00:57

Problem 33

Suppose that $p$ is a prime and $p \neq 2$. Let $a$ be a nonsquare in $\mathrm{GF}(p)$ - that is, $a$ does not have the form $b^{2}$ for any $b$ in $\mathrm{GF}(p)$. Show that $a$ is a nonsquare in $\mathrm{GF}\left(p^{n}\right)$ if $n$ is odd and that $a$ is a square in $\mathrm{GF}\left(p^{n}\right)$ if $n$ is even.

Trang Hoang
Trang Hoang
Numerade Educator
01:47

Problem 34

Let $f(x)$ be a cubic irreducible over $Z_{p}$, where $p$ is a prime. Prove that the splitting field of $f(x)$ over $Z_{p}$ has order $p^{3}$ or $p^{6}$.

Mohamed Raafat Mohamed
Mohamed Raafat Mohamed
Numerade Educator
03:55

Problem 35

Show that every element of $\mathrm{GF}\left(p^{n}\right)$ can be written in the form $a^{p}$ for some unique $a$ in $\mathrm{GF}\left(p^{n}\right)$.

Aman Gupta
Aman Gupta
Numerade Educator
02:08

Problem 36

Suppose that $F$ is a field of order 1024 and $F^{*}=\langle\alpha\rangle .$ List the elements of each subfield of $F$.

Lucas Finney
Lucas Finney
Numerade Educator
00:45

Problem 37

Suppose that $F$ is a field of order 125 and $F^{*}=\langle\alpha\rangle$. Show that $\alpha^{62}=-1$

James Kiss
James Kiss
Numerade Educator
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Problem 38

Show that no finite field is algebraically closed.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 39

Let $E$ be the splitting field of $f(x)=x^{p^{n}}-x$ over $Z_{p}$. Show that the set of zeros of $f(x)$ in $E$ is closed under addition, subtraction, multiplication, and division (by nonzero elements). (This exercise is referred to in the proof of Theorem 22.1.)

Nick Johnson
Nick Johnson
Numerade Educator
01:02

Problem 40

Suppose that $L$ and $K$ are subfields of $\mathrm{GF}\left(p^{n}\right)$. If $L$ has $p^{s}$ elements and $K$ has $p^{t}$ elements, how many elements does $L \cap K$ have?

AG
Ankit Gupta
Numerade Educator
01:59

Problem 41

Let $a$ be a non-zero element of $\mathrm{GF}\left(p^{n}\right)$. Prove that the number of solutions of $x^{p-1}=a$ is 0 or $p-1$.

James Chok
James Chok
Numerade Educator
13:07

Problem 42

Let $\alpha$ be a zero of an irreducible quadratic polynomial over $Z_{5}$. Prove that there are elements $a$ and $b$ in $Z_{5}[\alpha]$ such that $(3 \alpha+2)(a \alpha+b)$ $=4 \alpha+1$

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
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Problem 43

Show that a finite extension of a finite field is a simple extension.

Nick Johnson
Nick Johnson
Numerade Educator
01:08

Problem 44

Let $F$ be a finite field of order $q$ and let $a$ be a nonzero element in $F$. If $n$ divides $q-1$, prove that the equation $x^{n}=a$ has either no solutions in $F$ or $n$ distinct solutions in $F$.

Victor Salazar
Victor Salazar
Numerade Educator
01:05

Problem 45

Give an example to show that the mapping $a \rightarrow a^{p}$ need not be an automorphism for arbitrary fields of prime characteristic $p$.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:56

Problem 46

In the field $\mathrm{GF}\left(p^{n}\right)$, show that for every positive divisor $d$ of $n$, $x^{p^{n}}-x$ has an irreducible factor over $\mathrm{GF}(p)$ of degree $d .$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:58

Problem 47

Let $a$ be a primitive element for the field $\mathrm{GF}\left(p^{n}\right)$, where $p$ is an odd prime and $n$ is a positive integer. Find the smallest positive integer $k$ such that $a^{k}=p-1$

Trang Hoang
Trang Hoang
Numerade Educator
02:37

Problem 48

Let $a$ be a primitive element for the field $\mathrm{GF}\left(5^{n}\right)$, where $n$ is a positive integer. Find the smallest positive integer $k$ such that $a^{k}=2$.

Aman Gupta
Aman Gupta
Numerade Educator
03:43

Problem 49

Let $p$ be a prime such that $p \bmod 4=1$. How many elements of order 4 are in $\mathrm{GF}\left(p^{n}\right)^{*} ?$

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
03:43

Problem 50

Let $p$ be a prime such that $p \bmod 4=3$. How many elements of order 4 are in $\mathrm{GF}\left(p^{n}\right)^{*}$ ?

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator