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Contemporary Abstract Algebra

Joseph Gallian

Chapter 33

Cyclotomic Extensions - all with Video Answers

Educators

Chapter Questions

02:43

Problem 1

Determine the minimal polynomial for $\cos (\pi / 3)+i \sin (\pi / 3)$ over $Q$.

Aman Gupta
Aman Gupta
Numerade Educator
00:43

Problem 2

Factor $x^{12}-1$ as a product of irreducible polynomials over $Z$.

Sherrie Fenner
Sherrie Fenner
Numerade Educator
00:41

Problem 3

Factor $x^{8}-1$ as a product of irreducible polynomials over $Z_{2}, Z_{3}$, and $Z_{5}$.

Khanh Ha
Khanh Ha
Numerade Educator
03:04

Problem 4

$$
\text { For any } n>1, \text { prove that the sum of all the } n \text { th roots of unity is } 0 .
$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:21

Problem 5

$$
\begin{aligned}
&\text { For any } n>1, \text { prove that the product of the } n \text { th roots of unity is }\\
&(-1)^{n+1}
\end{aligned}
$$

Kevin Harmer
Kevin Harmer
Numerade Educator
03:08

Problem 6

Let $\omega$ be a primitive 12 th root of unity over $Q .$ Find the minimal polynomial for $\omega^{4}$ over $Q$.

Chris Trentman
Chris Trentman
Numerade Educator
04:15

Problem 7

Let $F$ be a finite extension of $Q$. Prove that there are only a finite number of roots of unity in $F$.

Donald Albin
Donald Albin
Numerade Educator
01:46

Problem 8

For any $n>1$, prove that the irreducible factorization over $Z$ of $x^{n-1}+x^{n-2}+\cdots+x+1$ is $\Pi \Phi_{d}(x)$, where the product runs over all positive divisors $d$ of $n$ greater than 1 .

Sanchit Gogia
Sanchit Gogia
Numerade Educator
02:57

Problem 9

If $2^{n}+1$ is prime for some $n \geq 1$, prove that $n$ is a power of $2 .$ (Primes of the form $2^{n}+1$ are called Fermat primes. $)$

Bryan Lynn
Bryan Lynn
Numerade Educator
01:42

Problem 10

Prove that $\Phi_{n}(0)=1$ for all $n>1$

Anna Jones
Anna Jones
Numerade Educator
01:22

Problem 11

Prove that if a field contains the $n$ th roots of unity for $n$ odd, then it also contains the $2 n$ th roots of unity.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
03:56

Problem 12

Let $m$ and $n$ be relatively prime positive integers. Prove that the splitting field of $x^{m n}-1$ over $Q$ is the same as the splitting field of $\left(x^{m}-1\right)\left(x^{n}-1\right)$ over $Q$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:01

Problem 13

$$
\text { Prove that } \Phi_{2 n}(x)=\Phi_{n}(-x) \text { for all odd integers } n>1 \text { . }
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:44

Problem 14

Prove that if $p$ is a prime and $k$ is a positive integer, then $\Phi_{p^{k}}(x)=$ $\Phi_{n}\left(x^{p^{k-1}}\right)$. Use this to find $\Phi_{8}(x)$ and $\Phi_{27}(x)$.

Aymara Gallardo
Aymara Gallardo
Numerade Educator
02:05

Problem 15

Prove the assertion made in the proof of Theorem $33.5$ that there exists a series of subgroups $H_{0} \subset H_{1} \subset \cdots \subset H_{t}$ with $\left|H_{i+1}: H_{i}\right|=2$ for $i=0,1,2, \ldots, t-1 .$ (This exercise is referred to in this chapter.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:41

Problem 16

Prove that $x^{9}-1$ and $x^{7}-1$ have isomorphic Galois groups over $Q$. (See Exercise 7 in Chapter 32 for the definition.)

Chris Trentman
Chris Trentman
Numerade Educator
04:02

Problem 17

$$
\begin{aligned}
&\text { Let } p \text { be a prime that does not divide } n . \text { Prove that } \Phi_{p n}(x)=\\
&\Phi_{n}\left(x^{p}\right) / \Phi_{n}(x)
\end{aligned}
$$

Mengchun Cai
Mengchun Cai
Numerade Educator
01:05

Problem 18

Prove that the Galois groups of $x^{10}-1$ and $x^{8}-1$ over $Q$ are not isomorphic.

Anthony Ramos
Anthony Ramos
Numerade Educator
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Problem 19

Let $E$ be the splitting field of $x^{5}-1$ over $Q$. Show that there is a unique field $K$ with the property that $Q \subset K \subset E$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 20

Let $E$ be the splitting field of $x^{6}-1$ over $Q .$ Show that there is no field $K$ with the property that $Q \subset K \subset E$.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 21

Let $E$ be the splitting field of $x^{6}-1$ over $Q .$ Show that there is no field $K$ with the property that $Q \subset K \subset E$.

Victor Salazar
Victor Salazar
Numerade Educator
01:08

Problem 22

$$
\begin{aligned}
&\text { Let } \omega=\cos (2 \pi / 15)-i \sin (2 \pi / 15) . \text { Find the three elements of }\\
&\operatorname{Gal}(Q(\omega) / Q) \text { of order } 2 .
\end{aligned}
$$

Hunza Gilgit
Hunza Gilgit
Numerade Educator