Contemporary Abstract Algebra

Joseph Gallian

Chapter 20 Extension Fields - all with Video Answers

Educators

Chapter Questions

00:10

Problem 1

Describe the elements of $Q(\sqrt[3]{5})$.

Erika Bustos

Numerade Educator

04:11

Problem 2

Show that $Q(\sqrt{2}, \sqrt{3})=Q(\sqrt{2}+\sqrt{3})$.

Ethan Somes

Numerade Educator

01:28

Problem 3

Find the splitting field of $x^{3}-1$ over $Q .$ Express your answer in the form $Q(a)$.

Kimberly Waterbury

Numerade Educator

01:08

Problem 4

Find the splitting field of $x^{4}+1$ over $Q$.

Victor Salazar

Numerade Educator

02:25

Problem 5

Find the splitting field of
$$
x^{4}+x^{2}+1=\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)
$$
over $Q$.

Ashley Volpe

Numerade Educator

01:06

Problem 6

Let $a, b \in \mathbf{R}$ with $b \neq 0$. Show that $\mathbf{R}(a+b i)=\mathbf{C}$.

Willis James

Numerade Educator

01:23

Problem 7

Find a polynomial $p(x)$ in $Q[x]$ such that $Q(\sqrt{1+\sqrt{5}})$ is ringisomorphic to $Q[x] /\langle p(x)\rangle$.

Maninder Singh

Numerade Educator

01:17

Let $F=Z_{2}$ and let $f(x)=x^{3}+x+1 \in F[x] .$ Suppose that $a$ is a zero of $f(x)$ in some extension of $F$. How many elements does $F(a)$ have? Express each member of $F(a)$ in terms of $a$. Write out a complete multiplication table for $F(a)$.

Adriano Chikande

Numerade Educator

00:50

Problem 9

Let $F(a)$ be the field described in Exercise $8 .$ Express each of $a^{5}$, $a^{-2}$, and $a^{100}$ in the form $c_{2} a^{2}+c_{1} a+c_{0}$

Mir Afzal

Numerade Educator

02:03

Problem 10

Let $F(a)$ be the field described in Exercise 8 . Show that $a^{2}$ and $a^{2}+a$ are zeros of $x^{3}+x+1$.

Kira Schwander

Numerade Educator

00:18

Problem 11

Describe the elements in $Q(\pi)$.

Linda Hand

Numerade Educator

00:53

Problem 12

Let $F=Q\left(\pi^{3}\right) .$ Find a basis for $F(\pi)$ over $F$.

Varsha Aggarwal

Numerade Educator

01:03

Problem 13

Write $x^{7}-x$ as a product of linear factors over $Z_{3}$. Do the same for $x^{10}-x$

John Irizar

Numerade Educator

00:36

Problem 14

Find all ring automorphisms of $Q(\sqrt[3]{5})$.

Ankit Gupta

Numerade Educator

01:42

Problem 15

Let $F$ be a field of characteristic $p$ and let $f(x)=x^{p}-a \in F[x]$. Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F$.

Frank Lin

Numerade Educator

01:39

Problem 16

Suppose that $\beta$ is a zero of $f(x)=x^{4}+x+1$ in some extension field $E$ of $Z_{2}$. Write $f(x)$ as a product of linear factors in $E[x]$.

Gregory Higby

Numerade Educator

05:30

Problem 17

Find $a, b, c$ in $\underline{Q}$ such that
$$
(1+\sqrt[3]{4}) /(2-\sqrt[3]{2})=a+b \sqrt[3]{2}+c \sqrt[3]{4}
$$
Note that such $a, b, c$ exist, since
$$
(1+\sqrt[3]{4}) /(2-\sqrt[3]{2}) \in Q(\sqrt[3]{2})=(a+b \sqrt[3]{2}+c \sqrt[3]{4} \mid a, b, c \in Q\}
$$

Sheryl Ezze

Numerade Educator

01:15

Problem 18

Express $(3+4 \sqrt{2})^{-1}$ in the form $a+b \sqrt{2}$, where $a, b \in Q$

Julie Silva

Numerade Educator

01:39

Problem 19

Show that $Q(4-i)=Q(1+i)$, where $i=\sqrt{-1}$.

Rakesh Kumar Sharma

Numerade Educator

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Problem 20

Let $F$ be a field, and let $a$ and $b$ belong to $F$ with $a \neq 0$. If $c$ belongs to some extension of $F$, prove that $F(c)=F(a c+b)$. (F "absorbs" its own elements.)

Nick Johnson

Numerade Educator

03:10

Problem 21

Let $f(x) \in F[x]$ and let $a \in F$. Show that $f(x)$ and $f(x+a)$ have the same splitting field over $F$.

Anurag Kumar

Numerade Educator

01:38

Problem 22

Recall that two polynomials $f(x)$ and $g(x)$ from $F[x]$ are said to be relatively prime if there is no polynomial of positive degree in $F[x]$ that divides both $f(x)$ and $g(x)$. Show that if $f(x)$ and $g(x)$ are relatively prime in $F[x]$, they are relatively prime in $K[x]$, where $K$ is any extension of $F$.

Archana Goyal

Numerade Educator

00:38

Problem 23

Determine all of the subfields of $Q(\sqrt{2})$.

Esraa Samir

Numerade Educator

02:33

Problem 24

Let $E$ be an extension of $F$ and let $a$ and $b$ belong to $E$. Prove that $F(a, b)=F(a)(b)=F(b)(a) .$

Mohamed Mohamed

Numerade Educator

00:11

Problem 25

Write $x^{3}+2 x+1$ as a product of linear polynomials over some extension field of $Z_{3}$.

Erika Bustos

Numerade Educator

00:15

Problem 26

Express $x^{8}-x$ as a product of irreducibles over $Z_{2}$.

Maninder Singh

Numerade Educator

01:05

Problem 27

Prove or disprove that $Q(\sqrt{3})$ and $Q(\sqrt{-3})$ are ring-isomorphic.

Anthony Ramos

Numerade Educator

04:29

Problem 28

For any prime $p$, find a field of characteristic $p$ that is not perfect.

Trang Hoang

Numerade Educator

02:44

Problem 29

If $\beta$ is a zero of $x^{2}+x+2$ over $Z_{5}$, find the other zero.

Brandon Collins

Numerade Educator

01:07

Problem 30

Show that $x^{4}+x+1$ over $Z_{2}$ does not have any multiple zeros in any extension field of $Z_{2}$.

Ankit Gupta

Numerade Educator

09:25

Problem 31

Show that $x^{21}+2 x^{8}+1$ does not have multiple zeros in any extension of $Z_{3}$.

Oswaldo Jiménez

Numerade Educator

00:56

Problem 32

Show that $x^{21}+2 x^{9}+1$ has multiple zeros in some extension of $Z_{3}$.

James Kiss

Numerade Educator

03:12

Problem 33

Let $F$ be a field of characteristic $p \neq 0 .$ Show that the polynomial $f(x)=x^{p^{n}}-x$ over $F$ has distinct zeros.

Karly Williams

Numerade Educator

01:09

Problem 34

Find the splitting field for $f(x)=\left(x^{2}+x+2\right)\left(x^{2}+2 x+2\right)$ over $Z_{3}[x]$. Write $f(x)$ as a product of linear factors.

James Kiss

Numerade Educator

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Problem 35

Let $F$ be a field and $E$ an extension field of $F$ that contains $a_{1}$, $a_{2}, \ldots, a_{n} .$ Prove that $F\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ is the intersection of all subfields of $E$ that contain $F$ and the set $\left\{a_{1}, a_{2}, \ldots, a_{n}\right\} .$ (This exercise is referred to in this chapter.)

Nick Johnson

Numerade Educator

02:38

Problem 36

Suppose that $a$ is algebraic over a field $F .$ Show that $a$ and $1+a^{-1}$ have the same degree over $F$.

Vikash Ranjan

Numerade Educator

02:23

Problem 37

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

Gregory Higby

Numerade Educator

04:07

Problem 38

Show that $Q(\sqrt{7}, i)$ is the splitting field for $x^{4}-6 x^{2}-7$.

Mengchun Cai

Numerade Educator

01:29

Problem 39

Let $p$ be a prime, $F=Z_{p}(t)$ (the field of quotients of the ring $Z_{p}[x]$ ), and $f(x)=x^{p}-t .$ Prove that $f(x)$ is irreducible over $F$ and has a multiple zero in $K=F[x] /\langle p-t\rangle$

Chris Trentman

Numerade Educator

03:11

Problem 40

Let $f(x)$ be an irreducible polynomial over a field $F$. Prove that the number of distinct zeros of $f(x)$ in a splitting field divides deg $f(x)$.

Abigail Darko

Numerade Educator

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Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

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Problem 40