Contemporary Abstract Algebra

Joseph Gallian

Chapter 32 An Introduction to Galois Theory - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $E$ be an extension field of $Q$. Show that any automorphism of $E$ acts as the identity on $Q$. (This exercise is referred to in this chapter.)

Anthony Ramos

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00:49

Problem 2

Determine the group of field automorphisms of GF(4).

Hast Aggarwal

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01:49

Problem 3

Let $E$ be an extension field of the field $F$. Show that the automorphism group of $E$ fixing $F$ is indeed a group. (This exercise is referred to in this chapter.)

Varsha Aggarwal

Numerade Educator

08:09

Given that the automorphism group of $Q(\sqrt{2}, \sqrt{5}, \sqrt{7})$ is isomorphic to $Z_{2} \oplus Z_{2} \oplus Z_{2}$, determine the number of subfields of $Q(\sqrt{2},$, $\sqrt{5}, \sqrt{7}$ ) that have degree 4 over $Q$.

Uma Kumari

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00:59

Problem 5

Let $E$ be an extension field of a field $F$ and let $H$ be a subgroup of Gal $(E / F)$. Show that the fixed field of $H$ is indeed a field. (This exercise is referred to in this chapter.)

Varsha Aggarwal

Numerade Educator

00:52

Problem 6

Let $E$ be the splitting field of $x^{4}+1$ over $Q$. Find $\operatorname{Gal}(E / Q)$. Find all subfields of $E$. Find the automorphisms of $E$ that have fixed fields $Q(\sqrt{2}), Q(\sqrt{-2})$, and $Q(i) .$ Is there an automorphism of $E$ whose fixed field is $Q$ ?

Sriram Soundarrajan

Numerade Educator

04:41

Problem 7

Let $f(x) \in F[x]$ and let the zeros of $f(x)$ be $a_{1}, a_{2}, \ldots, a_{n} .$ If $K=$ $F\left(a_{1}, a_{2}, \ldots, a_{n}\right)$, show that $\operatorname{Gal}(K / F)$ is isomorphic to a group of permutations of the $a_{i}$ 's. [When $K$ is the splitting field of $f(x)$ over $F$, the group $\operatorname{Gal}(K / F)$ is called the Galois group of $f(x) .]$

Harmender Singh Yadav

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00:59

Problem 8

Show that the Galois group of a polynomial of degree $n$ has order dividing $n !$.

Raj Bala

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01:57

Problem 9

Referring to Example 6 , show that the automorphism $\phi$ has order 6 . Show that $\omega+\omega^{-1}$ is fixed by $\phi^{3}$ and $\omega^{3}+\omega^{5}+\omega^{6}$ is fixed by $\phi^{2}$. (This exercise is referred to in this chapter.)

Wendi Zhao

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00:10

Problem 10

Let $E=Q(\sqrt{2}, \sqrt{5})$. What is the order of the group $\operatorname{Gal}(E / Q)$ ? What is the order of $\operatorname{Gal}(Q(\sqrt{10}) / Q)$ ?

Erika Bustos

Numerade Educator

06:05

Problem 11

Suppose that $F$ is a field of characteristic 0 and $E$ is the splitting field for some polynomial over $F$. If $\operatorname{Gal}(E / F)$ is isomorphic to $Z_{20} \oplus Z_{2}$, determine the number of subfields $L$ of $E$ there are such that $L$ contains $F$ and
a. $[L: F]=4$.
b. $[L: F]=25$.
c. $\operatorname{Gal}(E / L)$ is isomorphic to $Z_{5}$.

Jacob Fry

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00:56

Problem 12

Determine the Galois group of $x^{2}-10 x+21$ over Q. (See Exercise 7 for the definition).

Christopher Stanley

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00:59

Problem 13

Determine the Galois group of $x^{2}+9$ over $\mathbf{R}$. (See Exercise 7 for the definition).

Odera Egbuonu

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03:58

Problem 14

Suppose that $F$ is a field of characteristic 0 and $E$ is the splitting field for some polynomial over $F$. If $\operatorname{Gal}(E / F)$ is isomorphic to $D_{6}$ prove that there are exactly three fields $L$ such that $E \supseteq L \supseteq F$ and $[E: L]=6 .$

Anthony Ramos

Numerade Educator

02:35

Problem 15

Suppose that $E$ is the splitting field for some polynomial over $\mathrm{GF}(p)$. If $\operatorname{Gal}(E / \mathrm{GF}(p))=p^{6}$, how many fields are there strictly between $E$ and $\mathrm{GF}(p) ?$

Sahil Kumar

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02:20

Problem 16

Let $p$ be a prime. Suppose that $|\operatorname{Gal}(E / F)|=p^{2}$. Draw all possible subfield lattices for fields between $E$ and $F$.

Lucas Finney

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01:29

Problem 17

Suppose that $F$ is a field of characteristic 0 and $E$ is the splitting field for some polynomial over $F$. If $\operatorname{Gal}(E / F)$ is isomorphic to $A_{4}$, show that there is no subfield $K$ of $E$ such that $[K: F]=2$.

Chris Trentman

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00:47

Problem 18

Determine the Galois group of $x^{3}-1$ over $Q$ and $x^{3}-2$ over $Q$. (See Exercise 7 for the definition.)

Christopher Stanley

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01:26

Problem 19

Suppose that $K$ is the splitting field of some polynomial over a field $F$ of characteristic 0 . If $[K: F]=p^{2} q$, where $p$ and $q$ are distinct primes, show that $K$ has subfields $L_{1}, L_{2}$, and $L_{3}$ such that $\left[K: L_{1}\right]=p$, $\left[K: L_{2}\right]=p^{2}$, and $\left[K: L_{3}\right]=q$.

Ankit Gupta

Numerade Educator

01:29

Problem 20

Suppose that $E$ is the splitting field of some polynomial over a field $F$ of characteristic 0 . If $\operatorname{Gal}(E / F)$ is isomorphic to $D_{5}$, draw the subfield lattice for the fields between $E$ and $F$.

Chris Trentman

Numerade Educator

01:34

Problem 21

Suppose that $F \subset K \subset E$ are fields and $E$ is the splitting field of some polynomial in $F[x]$. Show, by means of an example, that $K$ need not be the splitting field of some polynomial in $F[x]$.

Jill Tolbert

Numerade Educator

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Problem 22

Suppose that $E$ is the splitting field of some polynomial over a field $F$ of characteristic $0 .$ If $[E: F]$ is finite, show that there is only a finite number of fields between $E$ and $F$.

Nick Johnson

Numerade Educator